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Citation: Ziai, Y. (1979). Statistical models of claim amount distributions in general insurance. (Unpublished Doctoral thesis, City University London)
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STATISTICAL M DELS OF CLAIM ANIOLM DISTRIBUTIONS
IN GENEERL\L INSUIW CE.
by
YOUSSEF ZIA?, M. Sc., A. I. A.
A Thesis Submitted for the Degree of Doctor of Philosophy
Actuarial Science Division
Dept: to nt of hiatheratics
The City University, London.
009,9 1 DZ q-76S/90
April 19'79
CONTENTS
ACKNOWLEDGEMENTS
ABSTRACT
QýAPTER 1 INTRODUCTION
1.1 Definition 1.2 Basic Problems of Insurance 1.3 Statistical Modelling 1.4 A Review of the Applications of Statistical
Distributions to Claim Amounts Data in General Insurance
1.5 Objectives and Outline of the Study 1.6 The Accidental Damage (AD) Data 1.7 Tables
CHAPTER 2 Tests for Goodness-of-Fit
2.1 Introducticn 2.2 The Chi-square Cocdness-of-Fit Test 2.3 Application Procedure for the Chi-square Test 2.4 The Total Expected Loss Statistic, T 2.5 Application Procedure for the T Statistic 2.6 The Kolmogorov-Smirnov Goodness-of-Pit Test 2.7 Comparison Between the Chi-square and the
Kolm gorov-Smirnov Goodness-of-Fit Tests 2.8 Application Procedure for the K-S Test 2.9 Other Goodness-of-Fit Tests 2.10 Tables
CQWPTER 3 The Lognormal Distribution
3.1 Introduction 3.2 Definition 3.3 Properties of the Two-Parameter Lognormal Distribution 3.4 Properties of the 3-Parameter Logformal Distribution 3.5 The Lognormal Distribution as a Model for Claim Amounts 3.6 Tests of Lognormality :- The 2-Parameter Case
3.6.1 The Graphical Test 3.6.2 The Skewness and Kurtosis Tests 3.6.3 Test in the (Bi, $2) Plane 3.6.4 Testing the Accidental Damage Data for
2-Parameter Lognormality
3.7 Estimation of the Parameters of the 2-Parameter Lognormal Distribution 3.7.1 The Method of Moments 3.7.2 The Approximate Maximum Likelihood Method 3.7,3 The Method of Quantiles 3., 7.4 The Method of Median and Coefficient of
Variation 3.7.5 The Graphical Method 3.7.6 The Method of Least Squares 3.7.7 The Mul tinomia1 >>aximuuri Likelihood Method 3.7.8 Measuring the Performance of Various
Methods of Cstimaticn
Pam
iv
V
1 1 2
4 4 8
11
19 19 21 27 29 33 33
37 38 41 42
43 43 45 46 49 50 51 51 52 53
54
59 60 61 62
63 64 65 67
70
1
Page 3.8 Application of the 2-Parameter Lognormal Model to the
Accidental Damage Data 7y
3.9 Prediction of the Claim Amount Distribution: - The 2-Parameter Lognormal Model 78
3.9.1 The Effects of Inflation on the Parameters of the Model 78
3.9.2 The Prediction Technique 79 3.9.3 Prediction for the AD Data 31
3.10 The 3-Parameter Lognormal Mode]. 84
3.11 Test of Lognormality: - The 3-Parameter Case 85
3.11.1 The Graphical Test 85 3.11.2 Testing the AD Data for 3-Parameter
Lognormality 86
3.12 Estimation of the Parameters of the 3-Parameter Lognormal Distribution 87
3.12.1 The Method of Least Squares 88 3.12.2 The Multinomial Maximum Likelihood Method 90
3.13 Application of the 3-Parameter Lognormal Model to the Al) Data
. 91
3.14 Prediction of the Claim Amount Distribution: - The 3-Parameter Lognormal Model 95
3.14.1 The Effects of Inflation on the Parameters of the Model 95
3.14.2 Prediction for the AD Data 96
3.15 Conclusions 98
3.16 Tables 99
CHAPTER 4 The Weibull Distribution 156
4.1 Introduction 156 4.2 Definition 157 4.3 Properties of the Weibull Distribution 158 4.4 The Graphical Test for the Weibull Distribution 160 4.5 The Weibull Graphical Test on the AD Data 163 4.6 Estij-, ation of the Parameters of the Weibull
Distribution 165 4.6.1 The Method of Least Squares 167 4.6.2 Multinomial Maximun Likelihood Method 169
4.7 Application of the Weibull Model to the AD Data 170 4.8 The Effects of Inflation on the Parameters
of the Weibull Model 175 4,9 Conclusions 176 4.10 Tables 177
CHAPTER 5 The Inverse Gaussian Distribution 186
5.1 Introduction 185 5.2 Definition 187 5.3 Properties of the Inverse Gaussian Distribution 189 5.4 Estimation of the Parc. rieters from Group: d Data 191
11
Pie
5.5 Application of the Inverse Gaussian Model to the AD Data 193
5.6 Prediction of the Claim Amount Distribution 197 5.6.1 The Effects of Inflation on the
Parameters of the Model 197 5.6.2 Prediction for the AD Data 198
5.7 Conclusions 199 5.8 Tables 200
CHAPTER 6 The Pareto Distribution 213
6.1 Introduction 213 6.2 Definition 213 6.3 Properties of the Pareto Distribution 214 6.4 The Graphical Test for the Pareto Distribution 215 6.5 The Pareto Graphical Test on the AD Data 216 6.6 Estimation of the Parameters of the Pareto
Distribution 218 6.7 The Effects of Inflation on the Parameters 220 6.8 Conclusions 0 221
CHAPTER 7 The Truncated Lognormal Distribution 223
7.1 Introduction 223 7.2 Definition and Some Properties of the Truncated
Lognormal Distribution 226 7.3 Estimation of Parameters 227 7.4 Applications to the AD Data 230 7.5 Conclusions 234 7.6 Tables 234
CHAPTER 8 The Gamma Distribution 250
8.1 Introduction 250 8.2 Definition and Some Properties of the Gamma
Distribution 253 8.3 Estimation of Parameters 257 8.4 Application of the Gamma Distribution to the AD
Square Root Data 258 8.5 The Effects of Inflation on the Parameters of the
Model 261 8.6 Prediction for the AD Square Root Data 261 8.7 Conclusions 262 8.8 Tables 264
CHAPTER 9 Summary of Conclusions 273
APPENDIX 278
REFE RENCES 313
111
ACKNOWLIDGEMEMS
I would like to express my gratitude to Professor Bernard Benjamin
for his helpful supervision, constructive criticism and kind
encouragement during the preparation of this thesis.
I am additionally grateful to Mr. Stewart Coutts for suggesting some
of the problems considered in this thesis and for his rigorous
criticism and constructive advice throughout.
The applications of. the theoretical aspects of this work are
demonstrated on actual data provided by a British insurance company
whose co-operation is gratefully acknowledged.
I would also like to thank the library staff of the Institute of
Actuaries and of the City University whose sympathetic assistance
aided my research.
Finally, I wish to record my appreciation of the work of
Miss Lancaster and Miss Williams in typing the manuscript.
iv
ABSTRACT
This work examines the following statistical distributions
as possible models for the distribution of claim amounts in general
insurance:
1- The lognormal
2- The Weibull
3- The inverse Gaussian (A new 3-parameter form is introduced)
4- The Pareto
5- The truncated lognormal as a model for large claim amounts)
6- The gamma (as a model for the distribution of the square root of claim amounts)
The properties. of the above distributions are investigated and various
methods of estimation of their parameters are explored. The method
of multinomial maxim mt likelihood for estimating the parameters is
favoured because data on claim amounts is generally in grouped
frequency format. To find these estimates a computing technique is
proposed which avoids solving a complicated set of non-linear
equations. A procedure which avoids solving non-linear equations
is also suggested for the least squares estimation of the 3-parameter
lognormal, 3-param_etor Weibull and the Pareto distribution of the
second kind. In order to show how the various methods work in
practice they are applied to an actual set of accidental damage claim
amounts. Goodness-of-fit tests are used to judge the agreement
between the model and sample valuos. The Chi-square and the
Kolmogorov-Sriirnov tests are reviewed and a new test statistic is
V
proposed which measures the overall agreement between the model
and sample values in monetary terms. The application procedures
for all these tests are described.
Inflation is likely to be the main cause of the increase in
the size of claim over time. Therefore, its effects on the
parameters of various models are examined. A method is suggested
for predicting the future distribution of claim amounts which
uses the parameters of a past model after being adjusted for
inflation. This predictive method is demonstrated on the
accidental damage data whenever a suitable model is found.
vi
7
CRU"TER 1
INI RODI TCTION'
1.1 Definition
General insurance' is defined as all classes of insurance
other than life insurance (or assurance). For a thorough comparison
between life and general insurance reference may be made to
Benjamin (1977). Fundamentally, general insurance has the following
distinguishing features as compared with life assurance:
1- Claim size not knoi%n in advance and often without limit.
2 -- Premium changes from year to year because contracts are
normally renewed annually.
3- More than one claim can arise under the same policy.
4- Volatility; large variances of both claim frequency
and claim amount especially the latter.
1.2 Basic Problems of Insurance
In insurance we are faced with the problems of charging
adequate premiums to cover a certain risk and the setting up of
reserves adequate to meet the cost of future claims with some margin
of profit. In life assurance, where the claim amount is i o:: n in
advance or can be determined by actuarial methods, these problems
have been solved by the use of the life table (which provides a
model for the probability of survival) in conjunction with discounting
functions. In general insurance, however, where the size of claim is
most usually not ie ot. n in advance, a prior estimate of the future
cost of claims is essential to the calculation of reserves. This
cost is a cc'mbinatic. -, of the frequency of claims and their size. it
- Sometimes referred to as Non-fife insurance.
I
is possible to treat each of these components of the cost separately
by collecting separate statistics on the frequency and size of claims.
In this work, we are concerned with the size of claims only. No claim
amount table exists for the calculation of the probability of
occurrence of a claim of a certain size.
1.3 Statistical Modelling _.
ý(''
It is ]mown that many random factors affect the size of the claim.
Statistical modelling is recognized as a rational tool of analysis for
problems in all areas of science and engineering where data variation
cannot be ignored. Therefore, it coninands considerable attention in
the solution of the basic insurance problems. Statistical modelling
assumes that there is a claim amount distribution underlying the
risk process. This distribution, once determined, enables us to
calculate the probability that if a claim occurs it will be not
greater than a: certain size. The shape of the claim amount distribution
is important in premiun determination and reserves calculations. As
Beard (1974) states, "a good theoretically derived model would be of
considerable help in dealing with practical estimation problems
arising frort the random fluctuations which arise in the relatively
small samples (in terms of the large claims) which commronly are all
that is available". In some classes of general insurance business
reinsurance is sought due to the likelihood of occurrence of very
large claims. In that case, the examination of the area (the
probability) under the upper tail of the distribution is necessary
for the insurance coy ; any's decision about 'retentioni' before
reinsurance. In motor insurance the policyholder may opt to pay
some first art of any on damage claim. ('voluntary excess') in
return. for a. reduction in the prerni. ta, i. The appropriate deduction,
1 -" That part of the risk which the insurance campany wishes to bear without help from the reinsures.
2
X-
in this case, may be calculated by examining the lower tail of the
claim amounts distribution. The essence of statistical modelling
approach is, therefore, to find a statistical distribution as a
model for the claim amounts experience of the particular class of
insurance in which we are interested. There are two stages involved.
At first we have to demonstrate from theoretical considerations of the
problem that a specific statistical distribution can model the claim
amounts experience. Sometimes it is not readily obvious how the
model could be theoretically derived and we have to start irnnediately,
from stage two. That is, by fitting our intuitive model to samples
of data from past experience and, perhaps, by using statistical
goodness-of-fit tests satisfy ourselves that the specific model
actually agrees with the claim amounts experience for our particular
class of business. Once we have found a model and gathered
sufficient Ielowledge about its parameters and how they behave with
respect to time, we will be able to use statistical techniques to
predict the distribution of the claim amounts arising during any
future period. This is our main objective in this approach.
In practice a solution to our problem, even starting from stage
two as mentioned above, is not easily obtainable because of many
undesirable factors such as insufficiency of the data, heterogeneity
of the data and data being only available in a certain form (e. g.
grouped or/and truncated). The empirical distributions of claim
amounts are by nature skewed to the right, i. e., there are many
small claims and much fewer larger claims. This leads us to the
examination of positively skewed statistical distributions as possible
models. It is also inrportant to study truncated distributions since
in practice, as in reinsurance, data on claims above a certain size
only may be available. Some of the models which have been more
often employed are referred to below.
3
1.4 A Review of the Applications of Statistical Distributions to Cl. a Amounts Data in General Insurance;
Studies have been made which involve fitting statistical
distributions to claim size data. Beard (1955) has fitted the
lognonnal to the American fire insurance property damage claim size
data. Beard 41957) gives a numerical example of the application of
the lognormal. and log-Pearson type I distributions to an experience
of fire ciairs in Denmark. In Beard (1964) a lognormal distribution
is fitted to a sample of settled. motor insurance claims, property
damage and liability claims mixed. Benckcrt (1962) fits lognormal
distributions to claims data from fire insurance, accident insurance
and motor third party insurance. Harding (1968) uses a truncated
lognorinal distribution as a model for the original amount of a claim
falling under the excess of loss reinsurance of motor business
contracts. Ferrara (1971) fits lognormal distributions to fire
insurance claim size data from several different industries. }3enckert
and Jung (1974) fit lognormal and Pareto distributions to data on
claims in fire insurance of dwelling houses reported between 1958
and 1969 by Swedish fire insurance companies. Finger (1976) uses the
lognormal distribution as a model for claim amounts in liability
insurance. Bickerstaff (1972) uses the lognormal as a model for the
distribution by size of auto collision claims.
The lognormal seems to be the most successfully used model in
general insurance. However, the above references do not deal extensively
with the various methods of estirating the parameters of the lognormal
distribution, and the efficiencies of these Methods, nor do they examine
some of the other skc: aed distributions.
1.5 Objectives and Cut line of the. Study
It vas, with the ubýýýýr. Yc; narl: s iii mind that the -present work was
4.
started. We study several skewed distributions as models for claim
amounts in general insurance. In order to show the applications of
these models in practice we apply their theoretical methods to a
set of real data fron motor insurance Accidental Damage (AD) claim
amounts. A description of the data will be provided in secticn
1.6.
Goodness-of-fit tests are used frequently in the present work
to examine the agreement between a model and sample values. Therefore,
in chapter 2 we consider several of these tests. The more widely
used Chi-square test is reviewed and a new test statistic is proposed
to supplement it. This statistic measures the overall agreement
between a model and sample values in monetary terms and, therefore,
its value can be easily interpreted. The importance of this
statistic is demonstrated in examining the agreement between
predicted and actual distributions since, in that case, it indicates
by how much we have overpredicted or underpredicted the total cost
of claims.
The Kolmogorov-Smirnov test for goodness-of-fit is a well established,
but less frequently used, test which is also reviewed in Chapter 2.
The application procedures for all three statistics are described and,
in later chapters, demonstrated on the AD data.
Because of the importance of the lognormal distribution in
general insurance it is extensively studied in Chapter 3. The two
and three parameter cases are considered. After defining the distribution
its properties are examined and a theoretical justification for the
emergence of the lognormal distribution as a model for claim amounts
is provided. Tests of 1ognormality and various estimation methods are
suggested for the two parameter. case when data only in grouped form
5
are available. These are later demonstrated on the AD data. A
computer simulation is carried out to measure the efficiency of
different methods of estimation. The multinomial maximum likelihood
(MML) method which is most suitable for grouped data is studied and,
with the wide availability of computers, a technique is suggested
for finding the estimates of the parameters. This iterative
procedure maximizes the loglikelihood function directly, via a
search for the optimum solution, starting from a set of initial
values. The effects of inflation on the parameters are then
considerecTand predictions are made for the distribution of claim
amounts, in a future period, by using different indices of wages
and prices. The agreement between the actual and predicted claim
amounts are tested by the goodness-of-fit tests described in
Chapter 2. The 3-parameter case is then studied. A method of
estimation which involves the least squares technique and a
search for the location parameter is suggested which avoids solving
non-linear equations. The method of IWL is also modified for the
3-parameter case. These methods are then applied to the AD data.
The effects of inflation on the parameters are studied and distributions
of claim amounts during future periods arc predicted.
In Chapter 4 the Weibull distribution which belongs to the
exponential family is studied. The two and three parameter cases are
examined, very much on the same lines as for the lognormal distribution.
The same methods of estimation as for the 3-parameter lognormal are
modified. for the 3-parameter. Weibull distribution.
Chapter 5 is devoted to the study of the inverse Gaussian (or inverse
normal) distribution. This is a skewed distribution with a shape
similar to the lognormal, the g aroma and the Weibull distributions .
Ü
Chhikara and Folks (1978) state that several sets of cmpirical data
wich they have investigated seem to be equally well represented
by the lognormal and the inverse Gaussian distributions. In the
absence of other considerations they recornend the use of the
inverse Gaussian distribution on the basis of the convenience of
working with it. Therefore, it seemed important to study and test
this distribution on the AD claims. Initially, the properties of
the distribution are examined and the methods of moments and ML
are suggested for the estimation of the parameters. These methods
are applied to the AD claims data and the effects of inflation are
investigated. We then introduce the 3-parameter version of this
distribution by bringing in a threshold parameter. No mention of
this case is made in chhikara and Folks (1978) or in Johnson and
Kotz (1970). We suggest using the MML method of estimation. This
method is then applied to the data. The effects of inflation on
the parameters are studied and predictions for future periods are
made and comapred with the actual experience by using goodness-of-
fit tests.
Chapter 6 looks at the Pareto distributions of the first and
second kinds. The properties and a graphical test are studied.
Various estimation methods are examined. In Chapter 7 we use the
method of MM to fit this distribution to the upper tail of the
AD claims data, i. e., claims greater than £600.
To study the tail of the claim amounts distribution, which
is of interest in reinsurance, we deal with the truncated lognormal
model in Chapter 7. The method of A'IL is developed and applied to
the truncated samples of AD data. The effects of truncation at
different points are studied.
7
In Chapter 8 we examine the ganuna distribution. Board (1978 -
personal co munication) suggests the gamma distribution as a model
for the distribution of the square root of the claim amount. We
would like to test this to see if by taking the square root of the
claim amount we can arrive at a better fitting model. For this
reason a different set of AD data was obtained where the claims
are grouped into different bands according to the square root of
their size. The data are better described in Chapter 8. The
properties of the gamm distribution are studied and the methods
of moments and rte, are suggested for the estimation of the parameters.
These are then applied to the data. The effects of inflation on the
parameters are studied and predictions for future periods arc made.
A final discussion and a sumnary of the findings of the study
are presented in Chapter 9.
The tables of results for every chapter are presented at the
end of the chapter.
All the computer programs used in this work have been written
by the author in Fortran 4 language and have been run via the
interactive terminals on the City University's ICL 1905E Computer.
The texts of the programs are presented in the Appendix.
1.6 The Accidental Damage (AD) Data
The important part of any data analysis is to have a reliable
set of data. Since we are looking for models of the distribution of
claim amounts we must be sure that the data used in the analysis has
been collected only from the experience of the particular class of
business we are investigating. In other words, the data must be
free from heterogeneity in every respect. In addition, a considerable
amount of data is required and we need to look at the experience
over several periods.
8
A miedium-sized, U. K., general insurance company provided its with
some data. of Accidental Damagel (AD), excluding windscreen, claim
amounts in respect of claims which occurred during certain periods
of time in the past. These periods are referred to as periods of
accident. T h-- portfolio from which these claims come was
comprehensively insured private cars. Data combined for all age
groups, vehicle groups, districts and type of use were available.
For claims iip to £570 we were given the number of open and settled
claims grouped in bands of £30, i. e. £i-30, £31-60, ........., £541-570.
For aniouiits greater than £571 details of the individual claims were
made available. After some investigation we grouped these in the
following bands:
£571 - £600, £601 - £700,701 - 5800, etc.
until the band containing the maximum claim amount (or amounts) was
reached.
Data was provided in respect of seven periods of accident. Each
period covers three months of the calendar year. Ile were given data
from the quarter starting on 1.10.1973 to the one ending on 30.6.1975.
For convenience sake we refer to the period, say, from 1.10.1973
to 31.12.1973 as '73/4th quarter'. There are, therefore, seven samples
of AD claim amounts each corresponding to one of the periods from
73/4th quarter to 75/2nd quarter. In each case the data had been collected
_ at least six months after the end of the period of accident. The
incurred but not reported (IBNR) claims are assumed to be so few as not
to present a problem. This is because experience (of the insurance
company) shows that almost all AD claims are reported and settled
within six months after the end of the period of accident. Zero
claims2 were not included in the data. 1- Damage to the policy holder's own vehicle. 2- Thee are those claims in respect Of s rich no pa ninety "l rn .
de by red or the insurance compan}' e: ' r ? 3eca1zse no na}ý, zcni is i eý, ui
because the insurance company recovers the cost fron: another insurer.
9
The relevant information for the construction of the Al. ) claim
amount samples were extracted from the computer print-outs of the
insurance company's claim files. These were then stored in data
subfiles on our computer.
Computer program P1 was written to print the frequency distribution
of a given sample of AD data. The sample cumulative distribution
function and various relevant sample statistics are also calculated
and printed out. The program was run with the AD samples and the
results are given in tables (1.1) to (1.7). In each table the
column headed by 'NCIII > LB. AMOUNT' gives the number of those claims
in the sairple whose amounts are greater than the amount given by the
lower boundary of each interval. From this information we can, for
example, see that in 73/4th quarter there were 89 claims with
amounts greater than £601. The tables show that the number of such
claims has increased over time. The total number of claims in each
sample is on average about 2600. For the calculation of the sample
moments we have assumed that in each interval all the claims are for
an amount equal to the mid-point of that interval. An inspection
of the insurance company's claim file showed that this assumption
was justified since the average amount of claim in each band was in
fact approximately equal to the mid-point of that claim amount interval.
For the calculation of the median and mode linear interpolations in
the claim amount intervals were used. From the tables we can see that the mean and the standard deviation of
the claim amounts have increased over time while the coefficient of
variation has rurw. ined quite stable at about 1.1. For each sample,
the frequency distribution, the coefficients of skewness and kurtosis
as well as the relative positions of the rode, the median and the mean
all indicate the skewness and sharp peakedness of the claim amounts
10
distribution. The sample statistics given in tables (1.1) to (1.7)
will be of use in later chapters.
1.7 Tables
11
Table (1.1)
ý`## 73/4TH QUARTER
AMOUNT £ NO. OF CLAIMS CUM. %%: NCLM > LB. AMOUNT 1- . 30 478 15.70 3045
31- 60 518 32.71 2567 61- 90 461 47.85 2049 91- 120 359 59.64 1588
121- 150 239 67.49 1229 151- 180 213 74.48 990 181- 210 148 79.34 777 211- 240 102 82.69 629 -241- 270 81 85.35 527 271- 300 58 117.26 446 301- 330 66 89.43 3118 331- 360 45 90.90 322 361- 3.90 39 92.111 277 391- 420 35 93.33 238 421- 450 34 94.45 203 451- 480 20 95.11 169 481- 510 29 96.06 149 511- 540 14 96.52 120 541'- 570 8 96.78 106
- 571- 600 9 97.08 98 601- Inn 29 98,03 R9 - 701- Ron 18 9E1.62 60 801- 900 20 99.28 42 901-1000 6 99.47 22
1001-1100 4 99.61 16 1101-1200 4 99.74 12 1201-1300 1 99.77 a 1301-1400 3 99.87 7 1401-1500 1 99.90 4 1501-1600 [i 99.90 3 160 1-1700 1" 99.93 3 1701-1800 0 99.93 2 1801-1900 0 99.93 2 1901-2000 1 99.97 2 2001-2100 0 99.97 1 2101-2200 0 99.97 1 2201-23(10 0 99.97 1 2301-2400 1 100.00 1
TOTAL NO. OF CLAIMS= 3045 MEAN CLAIM AMOUNT =150.36 STANDARD DEVIATION =175.48 COEFF. OF VARIATION= 1.17 MEDIAN ............ = 95.97 MODE .............. = 42. A7 SQRT61 =SKEWNESS = 3.50 82 ... _KURTOSIS = 24.20
-^ 107 G2 ................ =-0.5f6
G1 13 COEFF . OF SKEWNES: i FOn L0C; OF (, l.. AfA 0UN TS G02 15 EXCESS KURTOSIS FO LOG OF C: L. AfOOUNT`i.
12
Table (1.2)
### 24/15T QUARTER #####
AMOUNT E NO. OF CLAIMS CUM. Of, NCLM > LB. AMOUNT 1- 30 381 15.61 2441
31- 60 428 33.14 2060 61- 90 351 47.52 1632 91- 120 334 61.20 1281
121- 150 211 69.05 947 151- 180 133 75.30 736 181- 210 98 79.31 603 211- 240 82 82.67 505 241- 270 54 84.88 423 27.1- 300 52 87.01 369 301- 330 53 09.18 317 331- 360 36 90.66 264 361- 390 29 91.85 22B 391- 420 26 92.91 199 421- 450 22 93.81 173 451- 480 22 94.72 151 481- 510 17 95.41 129 511- 540 1() 95.02 112 541- 570 19 96.60 102 571- 600 4 96.76 83 601- 700 26 97.83 79 701- 800 21 98.69 53 801- 900 11 99.14 32 901-1000 10 99.55 21
1001-1100 5 99.75 11 1101-1200 2 99.84 6 1201-1300 1 99.88 4 1301-1400 2 99.96 3 1401-1500 0 99.96 1 1501-1600 0 99.96 1 1601-1700 0 99.96 1 1701-1800 1 100.00 1
TOTAL NO. OF CLAIMS= 2441 MEAN CLAIM AMOUNT =149.85 STANDARD DEVIATION =172.20 COEFF. OF VARIATION= 1.15 MEDIAN ............ = 95.93 MODE .............. = 41.87 SQRTB I -SKEWNESS = 2.83 B2 ... =KURTOSIS = 14.24 G1 .............. .. =-0.076 G2 ................ =-0.503
G1 IS COEFF. OF SKEWNESS VOR LOG OF CL. AMOUNTS G2 IS EXCESS KUR70SIS FOR LOG OF CL. AMOUNTS.
13
Table (1.3)
#*### 74/2ND QUARTER -f*#**
AMOUNT NO. OF CLAIMS CUM. % NCLM > LO. 1- 30 351 14.73 2383
31- 60 380 30.68 2032 61- 90 382 46.71 1652 91- 120 295 59.09 1270
121- 150 211 67.94 975 151- 180 142 73.90 764 181- 210 114 7A. 68 622 211- 240 101 82.92 508 241- 270 57 85.31 407 271- 3(10 51 87.45 350 301- 330 39 89.09 299 331- 360 36 90.60 260 361- 390 25 91.65 224 391- 420 24 92.66 199 421- 450 27 93.79 175 451- 480 18 94.54 148 181- 510 21 95.43 130 511- 540 17 96.14 109 541- 570 12 96.64 92 571- 600 11 97.10 80 601- 700 30 98.36 69 701- 800 13 9E3.91 39 801- 900 11 99.37 26 901-1000 4 99.54 15
1001-1100 7 99.83 11 1101-1200 0 99.83 4 1201-1300 1 99.87 4 1301-1400 1 99.92 3 1401-1500 0 99.92 2 1501-1600 1 99.96 2 1601-1700 0 99.96 1 1701-1800 0 99.96 1 1801-1900 1 100.00 1
TOTAL NO. OF CLAIMS= 23R3 MEAN CLAIM AMOUNT =151.57 STANDARD DEVIATION =167.98 COEFF. OF VARIATION= 1.11 MEDIAN ............ = 98.4A MODE .............. = 61.17 SQRTB 1 =SKEWNESS =2 . 88 62 ... =KURTOSIS = 15.91 G1 ................ =-0.14 G2 ................ =--f). nb4
G1 IS COEFF . OF SKE WNES S) - Fon LO1Ga 0F CL . j«% MCUN'i'S G2 IS FOP LOG OF CL. AMOUNTS.
AMOUNT
14
Table (1.4)
*3*## /4/3RD 1UARTE9 #####
AMOUNT E NO. OF CLAIMS CUM. % NCLM > LB. AMOUNT 1- 30 362 12.93 2799
31- 60 427 28.19 2437 61- 90 383 41.87 2010 91- 120 356 54.59 1627
121- 150 283 64.70 1271 151- 180 194 71.63 9138 181- 210 137 76.53 794 211- 240 97 79.99 657 241- 270 86 83.07 560 271- 300 71 85.60 474 301- 330 64 87.89 403 331- 360 45 89.50 339 361- 390 44 91.07 294 391- 420 25 91.96 250 421- 450 26 92.89 225 451- 480 22 93.68 199 481- 510 25 94.57 177 511- 540 14 95.07 152 541- 570 14 95.57 138 571- 600 17 96.18 124
., _ 601- 700 32 97.32 107 701- 800 34 98.54 75 801- 900 17 99.14 41 901-1000 4 99.29 24
1001-1100 9 99.61 20 1101-1200 4 99.75 11 1201-1300 0 99.75 7 1301-1l00 1 99.79 7
1401-1500 3 99.89 6 1501-1600 0 99.89 3 1601-1700 1 99.93 3 1701-1800 0 99.93 2 1801-1900 0 99.93 2 1901-2000 0 99.93 2 2001-2100 0 99.93 2 2101-2200 1 99.96 2 2201-2300 0 99.96 1 2301-2400 0 99.96 1 2401-2500 0 99.96 1 2501-2600 1 100.00 1
TOTAL NO. OF CLAIMS= 2 MEAN CLAIM AMOUNT =166.13 STANDARD DEVIATION =1RR. 70 COEFF. OF VARIATION= 1.14 MEDIAN ............ =109.6 7 MODE .............. _ 48.39 SORTOI -SKEWNESS = 3.42 P. 2 ... =KUPPTOSIS = 24.04 G1 ................ =-0.179 G2 ................
G1 IS COEFF. OF SKEW'NE iii FOR LOG OF CL. AMOUNTS G2 IS EXCESS KURTOSIS FOR LOG OF CL, AMOUNTS.
Table (1.5)
##### 74/4TH QUARTER #####
AMOUNT £ NO. OF CLAIMS CUM. % NCLM > LB. AMOUNT 1- 30 394 12.86 3064
31- 60 452 27.61 2670 61- 90 426 41.51 221A 91- 120 348 52.87 1792
121- 150 272 61.75 1444 151- 180 219 68.90 1172 181- 210 154 73.92 953 211- 240 124 77.97 799 241- 270 105 81.40 675 271- 300 78 83.94 570 301- 330 75 (36.39 492 331- 360 58 8h. 28 417 3611- 390 55 90.08 359 391- 420 29 91.02 304 421- 450 43 92.43 275 451- 480 24 93.21 232 481- 510 22 93.93 208 51 1'- 540 24 9A. 71 I A6 541- 570 19 95.33 162 571- 600 14 95.79 143
- 601- 700 42 97.16 129 701- 800 28 98.07 87 801- 900 20 98.73 59 901-1000 17 99.28 39
1001-1100 8 99.54 22 1101-1200 5 99.71 14 1201-1300 1 99.74 9 1301-1400 5 99.90 8 1401-1500 0 99.90 3 1501-1600 1 99.93 3 1601-1700 0 99.93 2 1701-1800 0 99.93 2 1801-1900 0 99.93 2 1901-2000 0 99.93 2 2001-2100 1 99.97 2 2101-2200 0 99.97 1 2201-2300 0 99.97 1 2301-2400 1 100.00 1
TOTAL NO. OF CLAIMS= 3064 MEAN CLAIM AMOUNT =174.19 STANDARD DEVIATION =194.03 COEFF. OF VARIATION= 1.11 MEDIAN ............ =112.91 MODE .............. = 51.21 SQFTB1 =SKEWNESS = 2.97 82 ... =KURTOSIS 17.65 Gi ................ =-0.199 Cc' ................ =-0.429
31 IS COEFF. OF SKEWNESS FOR LOG OF CL. AMOUNTS G2 IS EXCESS KURTOSIS FOR LOG OF CL. AMOUNTS.
16
75/1ST QUARTER *-W ** *
AMOUNT £ NO. OF CLAIMS CUV" .% NCLkli > LU . 1- 30 324 12.43 2607
31- 60 31J 27.27 22133 61- 90 345 40.51 1896 91- 120 289 51.59 1551
121- 150 25D 61.30 1262 151- 1130 187 68.47 1009 181- 210 1313 73.76 822 211- 240 114 78.14 684 241- 270 93 91.70 570 271- 300 67 84.27 477 301- 330 63 136.69 410 331- 360 44 138.38 347 361- 390 44 90.07 303 391- 420 35 91.41 259 421- 450 25 92.37 224 451- 480 26 93.36 199 4131- 510 1R 94.05 173 511- 540 113 94.74 155 541- 570 22 95.59 137 571- 600 17 96.24 115 601- 700 39 97.74 98 701- 800 19 913.47 59 1301- 900 18 99.16 40 901-1000 12 99.62 22
1001-1100 3 99.73 10 1101-1200 1 99.71 7
. 1201-1300 1 99.81 6 1301-1400 0 99.81 5 1401-1500 1 99.85 5 1501-1600 0 99.85 4 1601-1700 1 99.68 4 1701-1800 0 99.88 3 1801-1900 0 99.88 3 1901-2000 0 99.88 3 2001-2100 1 99.92 3 2101-2200 1 99.96 2 2201-2300 0 99.96 1 2301-2400 0 99.96 1 2401-2500 0 99.96 1 2501-2600 0 99.96 1 2601-2700 0 99.96 1 2701-21100 0 99.96 1 2801-2900 0 99.96 1 2901-3000 0 99.96 1 3001-3100 0 99.96 1 3101-3200 0 99.96 1 3201-3300 0 99.96 1 3301-3400 0 99.96 1 3401-3500 0 99.96 1 3501-3600. 1 100.00 1
TOTAL NO. OF CLAIMS= 2607 MEAN CLAIM AMOUNT =173.40 STANDARD DEVIATION =195.30
AMOUNT
COEFF. OF VAf 1ATION= 1.13 MEDIAN ............ =116.19 MODE ..............: 48.50 Sran'rr31 -SKEWNESS /--' . 39 132 ... . =: KURTOFI = 18.4 -1 17 G1 ................ =-0.23) G2 ....... =-0 . 394
Tal P (1ý?,
# #'" 75/2ND QUARTER #9--*#
AMOUNT £ NO. OF CLAIMS CU", 1. % NCLM > LB. AMOUNT 1- 30 302 12.10 2495
31- 60 374 27.09 2193 61- 90 332 40.40 1819 91- 120 27? 51.50 1487
121- 150 235 60.92 1210 1'51- 180 167 6P, . 42 975 181- 210 122 73.31 7138 211- 240 110 77.72 666 241- 270 80 60.92 556 271- 300 72 83.81 476 301- 330 47 65.69 1104 331- 360 39 87.25 357 361- 39n 40 88.06 318 391- 420 3A 90.38 278 421- 450 29 91.54 240 451- 480 21 92.38 211 481- 510 30 93.59 190 511- 540 19 94.35 160 541- 570 17 95.0.3 141 571- 600 11 95.47 124 601- 700 36 96.91 113 701- 800 22 97.80 77 801- 900 22 98.68 55 901-1 [Ioo 11 99.12 33
1001-11(10 4 99.28 22 1101-1200 3 99.40 18 1201-1300 3 99.52 15 1301-1400 6 99.76 12 1401-1500 2 99.84 6 1501-1600 1 99.88 4 1601-1700 0 99.88 3 1701-1800 1 99.92 3 1801-1900 0 99.92 2 1901-2000 1 99.96 2 2001-2100 0 99.96 1 2101-2200 0 99.96 1 2201-2300 1 100.00 1
TOTAL NO. OF CLAIMS= 2495 MEAN CLAIM AMOUNT =180.08 STANDARD DE1/IATI_ON =204.79 COEFF. OF. VARIATION= 1.14 MEDIAN ............ =116.44 MODE .............. = 49.45 SQRTF31 =SKEWNESS = 3.07
02 ... =KURTOSIS = 17.70 Cl ................. -0.1R4 G2 ................ =-0.376
GI IS COEFF. OF FKF'WNES ; FOR LOG OF CL. AMOUNTS G2 IS EXCESS KURTO IS FOR LOG OF CL. AMOUNTS.
1$
CHAPTER 2
Tests for Goodness-of-Fit
2.1 Introduction
The natural order of any statistical analysis involving the
fitting of a theoretical distribution to a set of sample values is
to fit the theoretical model first and then to test its agreement
with the observed distribution of the sai le values. This problem
of testing "the goodness-of-fit" (i: e. the adherence of the model
to the data) will arise at various stages of the present work when
we consider different statistical distributions as models for the
distribution of claim amounts. Before considering the fitting
methods for various distributions it is deem: d convenient to devote
one chapter to the study of the theoretical bases and application
procedures of some goodness-of-fit tests.
Goodness-of-fit tests are performed to examine the agreement
between the theoretical distribution of a random variable and its
empirical distribution represented by a set of sample values. In
other words, if xl, x2,..., xn are independent observations of a
random variable X (for instance the claim amount) with an unknown
distribution function F(Lx), then we are required to test the null
hypothesis that
H: F (x) = F0 (x)
where Fo(x) is some particular distribution function. Any test of
110 is called a. test of fit. Hypotheses of fit, Ho, may be
classified as simple or composite. H0 is a sigle hypothesis if
it specifies the values of all the parameters of F0(x). If the
values of none of the parameters or only of some of them are specified
by the nufl hypothesis týýe?: is is called a composite }iypothc. sis.
19
The reason for formulating the null hypothesis of a test of
fit in terms of the distribution function is that the parametric
hypothesis testing methods do not provide the means of testing
whether observations come from a particular distribution with
unspecified paramieters. In addition, by our intuition we expect
that the distribution of sample observations would closely
approximate the true distribution (Kendall and Stuart (1973)).
It is in this sense that a goodness of fit test is a measure of
the discrepancy between the sample and theoretical distribution
functions. Savage (1953) characterizes a goodness of fit test by
the following four properties:
1- It is defined for samples from some large class of distributions.
2- The null hypothesis is either some specified distribution or
a class of distributions of which the functional form is
known.
3- For all null hypotheses, the test statistic used has the
same distribution (at least asymptotically).
4- The test is consistentl.
In our work a test of goodness of fit will be required in
two circumstances:
(i) Fitting and Testing
Let us assume that we have a sample of data representing the observed distribution of the claim amounts. We can
postulate the form of the population distribution of this variable and use one of the appropriate estimation techniques, for that particular distribution, to estimate its parameters and, thus specify it completely. We will
1- A test of hypothesis 1-I0 against a class of alternatives Hl- is said to be consistent if, c; en any member of Ht holds, the probability of rejecting Ho tends to 1 as scinpl e size (s) tends to infinity.
20
then need to test how well the theoretical and observed distributions agree. A good agreement will be taken as
evidence that the assuned family of distributions is the
correct form for the distribution of claim amounts. The null hypothesis in this case is of the composite type.
Pr', d. ýction and Testing,
We may predict a theoretical distribution, whose form and
parameters are-completely specified, as the distribution
of claim amounts in a particular period of accident
occurrence. If a set of data representing the observed distribution for the same period already exists, we will be
able to test how well the predicted and actual distributions
agree. In this situation the null hypothesis of test is of
a simple type. Evidence of a good fit can be used to
recommend a prediction technique and support the assumption
about the theoretical model.
In. this chapter several goodness of fit tests will be studied.
In Section 2.2 the Chi-square goodness of fit test will be dealt
with in detail. To supplement the Chi-square test, and to avoid
some of its shortcomings, a test statistic based on the weighted
sum of the actual minus expected number of claims, in different intervals,
will be proposed in section 2.4. The Kolmogorov-Smirnov test of
goodness of fit, which is generally believed to be more powerful than
the Chi-square test, will be studied in detail in section 2.6. Two
other test statistics will be mentioned in section 2.9 but because
they are not applicable to grouped data we will not study them in
detail.
2.2 The Chi-scauare Goodness-of-Fit Test
This was the first goodness of fit test and it was introduced
by Karl Pearson in 1900.
Let us assizie that we have a sample of n independent observations of
a random variable X, with distribution function F (x) . P'earson's
test involves grouping the observations into, say, k mutually
2' 1
exclusive categories such that nl be the number of observations in k
category i and n=E ni i=1
If the population distribution is completely specified by the null
hypothesis as F(x) = Fo(x) (i. e. the form and all the parameters of
Fo(x) are knouzi) and we assume that Iö is true, then the probability,
poi, of a random observation falling into any category i can be
calculated. If poi is multiplied by the total number of
observations, n, we will find the expected frequency of class i, say noi'
under the hypothesis H0. Apart from sampling variation there should
be close agreement between ni and nof, the observed and expected frequencies respectively. The Chi-square goodness of fit test
provides a probability basis for computing and deciding whether the
discrepancy is too large to have occurred by chance. The test
statistic, proposed by Pearson, is;
2 X2 =E
(ni noi) i=1 not
Large values of X2 indicate an overall lack of agreement between the
observed and expected distributions. The null hypothesis which
resulted in Hors should, therefore, be rejected for large values of
x2.
If when Ho is assumed true the sampling distribution of a test
statistic is known and tractable, tables of its percentage points
can be constructed. The sampling distribution of X2 is very
complicated when the sample size is finite, but Cramer (1946) has
shown that its limiting distribution, under Ho, is approximately
Chi-square, X2, with k-i degrees of freedom (we assured F0(x)
22
as completely specified).
In some situations the null hypothesis is composite such
that usually only the form of the postulated distribution is laiown
but not any, or some, of its parameters. If these parameters have
to be estimated from the sample data, then the limiting distribution
of X2 may depend on the method of estimation. With a poor method
of estimation X2 may frequently have a large value even if the
theory is correct (Cochran (1952)). In a general proof of the
distribution of X2 the method of estimation must be asserted. A
method that yields those values of the parameters which minimize
X2 (mininun Chi-square method) may seem the most suitable. Fisher
(1924) has shown that in the limit in large samples this method
becomes equivalent to the method of multinomial maxi mm likelihood.
The following theorem due to R. A. Fisher, states the principal
theoretical result for the distribution of the Chi-square test
" statistic when parameters of the distribution of X under the null
hypothesis, Fo(x), are estimated from the sample data.
Theorem
If Fo(x), whose form is known, has r ui iown parameters, and
if the corresponding multinomial maximum likelihood estimates
are substituted for the unimovm parameters, then X2 is
distributed, in the lint, as the Chi-square, x2,
distribution with k-r-1 degrees of freedom.
(A proof of this can be found in Cramer (1946)).
Cochran (1952) states that any efficient method of estimation gives
estimates which in the limit become identical with the maximtnn
likelihood estimates. Thus, the Chi-square distribution with the
appropriate reduction in the degrees of freedom is valid, as the
23
distribution of X2 statistic, for any efficient method of
estimation. If fully efficient ordinary maximum likelihood
estimators are used, then X2 does not have an as)mptotic
Chi-square distribution. There will be a partial recovery of
the r degrees of freedom lost by the multinomial inaximi
likelihood estimators and the distribution of the Chi-square
statistics, X2, will be bounded between a x2k-1 and a X2k-r-1 variable.
Therefore the critical values should be adjusted upwards. As
k becomes large x2k-1 and x2k-r-1 become so close together that
the difference can be ignored (Kendall and Stuart (1973)).
If individual observations are available in the sample, and
the null hypothesis is that they follow a continuous density function,
then the investigator must first group the observations into
different mutually exclusive classes. Cochran (1952) mentions that
the investigator has the choice of both the number of classes and
the division points between them, but that his choice will affect
the sensitivity of the test. According to Kendall and Stuart
(1973) the whole asymptotic theory of the Chi-square test is
valid as long as the k'classes into which the observations are
grouped are determined without reference to the observations
because there has been no provision in the theory for the class
boundaries themselves being random variables. A rule suggested
by Mann and Wald (1942) and by Gumbel (1943) is to choose the
classes so that the expected frequencies are all equal to n/k
where k, the nwnber of classes, is assumed given. Mann and Wald
(1942) have developed a technique for finding the optimum nuTa)cr of
classes for any sample size n such that the power of the test is
never less than 21. The problems of the choice of the number of
classes and class boundaries do not arise i hcn only a sample of
24
already grouped observations is available. Such is the case with
our Accidental Damage data and the above problems are therefore
pursued no further here.
In the derivation of the sampling distribution of the Chi-square
test statistic, a multinomial distribution is approximated by a
multinormal distribution (see Cramer (1946) or Kendall and Stuart (1973)).
When the nu:. ber of classes is large, and the expected frequencies are
small, this approximation may not be satisfactory. It has been
suggested by some authors that the expected frequency in any interval
should not be less than S. For any expected frequency less than
5, the usual procedure is to pool the adjacent classes together until
this condition is removed. The number of degrees of freedom should
then be calculated on the basis of the number of classes actually
used, after pooling together, in the calculation of X2. Since the
discrepancy between an observed and a postulated distribution is
often most apparent in the tails, the sensitivity of the Chi-square
test is likely to be decreased by excessive pooling at the tails
(Cochran (1952)). Consequently the rule of minimum expected
frequency of 5 should not be considered as inflexible. Cochran
(1942) has shown that there is little disturbance at the 5% level
when a single expected frequency is as low as 1. At 1% level the
same is true if the degrees of freedom of the Chi-square distribution
are greater than 6. He states that two expected frequencies as low
as 1 may be allowed with negligible disturbance to the 5% level.
The Oil-square goodness-of-fit test is commonly used when there
is no clear alternative hypothesis. This necessarily precludes the
computation of power. If there exists an alternative hypothesis,
then the distribution of X2 under the alternative hypothesis is
asymptotically a "non-central x2" with k-r-1 degrees of freedom
25 1
.ý
(where r pararieters have been estimated by the multinomial maximum
likelihood estimators) and a non-centrality parameter which depends
on the sample size. For the method of calculation of the limiting
power function of the test see Kendall and Stuart (1973).
The dependence of the power of the Chi-square test on the sample size
is a weakness of this test, since with a small sample, an alternative
hypothesis which has a large departure from the null hypothesis, Ho,
may have a small probability of yielding a significant value of
the test statistic. On the other hand, for a large sample, rather
small and unimportant departures from the null hypothesis are likely
to yield a significant value of the test statistic.
Xe are two major shortcomings of the Chi-square test. lVhen
testing the goodness-of-fit of a continuous distribution, grouping the
observations into classes necessarily implies the loss of information
by such grouping. Another shortcoming is that the X2 test statistic
is based on the squares of the deviations between the actual and
expected frequencies. This implies that the Chi-square test will not
be sensitive to the pattern of signs of the deviations.
? The Chi-square goodness-of-fit test is applicable to situations
in tthich the alternative hypotheses are expressed in vague and general
terms. Its main advantage is that when the hypothesized distribution
is not completely specified the test can still be performed, in the
same way as for a simple null hypothesis, simply by replacing the
unknown parameters with efficient estimators and reducing the degrees
of freedom. by the nmiber of estimated parameters.
It was mentioned that the Chi-square goodness-of-fit test ignores
the signs of deviations of the observed frequencies from the expected
ones. It is, therefore, sometimes informative to examine the pattern
of the signs of deviations. David (1947) has shown that if the null
hypothesis is simple then all patterns of signs are equiprobable. 26
Therefore, when 11 is simple, if the hypothetical. distribution is
the true population distribution, we expect the signs of deviations
to have a random pattern rather than form a few clusters of
deviations of the same signs. To test the randomness pattern of
the signs, it is possible to use the "runs"' test (see, for instance,
Bury (1975)). A simpler alternative method is to compare the number
of changes of signs of deviations with the number of non-changes. z
These two numbers should be approximately equal, provided that a
large number of deviations exists and the pattern of signs is
random (Benjamin and Haycocks(1970)).
For. a composite null hypothesis, where all the parameters have to
be estimated from the sample, Fraser (1950) has shown that all
patterns of signs of deviations are not equiprobable. Therefore,
the "runs" test for randomness or the simple comparison of the
number of changes and non-changes of signs cannot be applied in
' the case of a composite null hypothesis.
2.3 App ication Procedure for the Chi-square Test
The frequency distributions of our AD claim amounts samples,
which were presented in Chapter 1, are skewed to the right and in
grouped form. In this work, the Chi-square goodness-of-fit tests
will be performed in situations (i) and (ii) mentioned in section 2.1.
If estimates of the parameters of a hypothesized distribution are
required, they will be calculated from the grouped data, as given,
without any amalgamation of the intervals. there necessary, computer
programs will be written to provide result tables, giving the actual,
A, and the expected, E, frequencies as well. as deviations, A-E, for
1- A "r&", in this case, is defined as a sequence of consecutive plus signs or minus signs.
2- In the sequence ++T--+ there are two changes and three non-changes of signs.
27
every interval. The value of for any interval in which E
the expected frequency is greater than or equal to 5, as well as
the total of such values are also printed. For other intervals,
i. e. where E<5, enough intervals will have to be pooled together
to remove this condition. The values of LA-Fý2 for the pooled E
intervals will then have to be calculated, on an electronic desk
calculator, and added to the total already given in the table.
The number of degrees of freedom is calculated on the basis of the
number of intervals actually used in the calculation of the final
value of X2. Considering that the number of intervals used in the
calculation of X2 is always more than 20, the value of X2 is
expected to be large. Because of the skewness of the data the
major contributions to the value of X2 are from the intervals in
the lower tail of the distribution. The number of claims in the
upper tail of the distribution, for instance, claims of amounts
greater than £1200, is very small. IT erefore the contributions to
the total X2 from, at most one or two, pooled intervals in the upper
tail will be rather small, when compared with the value of X2, and
may be ignored in many instances. This means that the total of (A-E) 2
E values are calculated by the computer, for the intervals with
an expected frequency of greater than or equal to 5, may be safely
taken as the value of the Chi-square statistic: However, for the
calculation of the degrees of freedom, the number of pooled intervals
will be taken into account. The above procedure saves us many
unnecessary calculations on the desk calculator.
Let us now assume that x is the value of our test statistic and
v is its relevant n rber of degrees of freedom. We look up a table
of ci iulati. ve percentage points of the Chi-square distribution,
28
with v degrees of freedom, and note the probability, P, that a 2
xv random variable will exceed the value of our calculated X2.
In other words we find P such that
P= Pr (X2 ^ X2) v
We classify the result of the test, according to different values
of P, as follows:
If (i) P30.05
or (ii)0.01, P< 0.05
or (iii) 0.001 5P0.01
or (iv) p<0.001
then the difference between the observed and postulated distributions,
tinder the null hypothesis, based on the given sample is respectively:
(i) Not Significant
(ii) Almost Significant
Significant
(iv) Highly Significant
The above arbitrary classification is based on a 5% significance level.
A different level of significance, say 10%, may be used if stronger
confidence is required from the test.
2 .4 The Total Expected Loss Statistic, T
It is not just the sign of the deviations that are considered
important but their magnitude needs some attention too. It is generally
expected that for a good fit the mag itude of the deviations should be
small. For a random variable such as the claim amount whose distribution
is skewed to the right, the frcqucncies in the intervals of the lower
29
tail of the sample histogram will be much greater than those in
the upper tail intervals. Therefore, the magnitude of the
deviations in the lower tail intervals will be generally greater
than those in the upper tail. On the other hand, the claim amounts
are larger in the upper tail than in the lower. Therefore, it is
not very informative merely to look at the magnitude of the
deviations. The above can perhaps be explained better by providing
an example. Suppose that the (Actual-Expected) frequency in the
£1-130 interval is equal to +100. In money terms this difference
is on average equal to
100 x £15.5 = 11550
where 115.5 is the average claim amount in that interval. This is,
in monetary terms, equivalent to a deviation of only 1 in the
11501 - £1600 interval. We, therefore, suggest looking at the
weighted deviation for each interval where the weight is the average
amount of claim in the interval (the examination of our data showed
that the mid-point of each interval is approximately equal to the
average claim amount in that interval). We can also define the test
statistic T as"the sum of the weighted deviations, i. e.
k T
ii £i(ni-noi)
where fi is the mid-point of interval i, in
ni is the actual frequencyLinterval i,
not is the expected frequency in interval i,
and k is the number of intervals.
This statistic is a measure of overall agreement, in monetary terns,
between a hypothesized distribution and the actual sample values. For
a good fit we expect the value of T to be small. Every component of T,
such as 30
ti (n1-nol)
indicates, in money terms, a difference (or a loss) in interval i
which we should expect to find between the observed distribution
of claim amounts and its hypothesized distribution under the null
hypothesis. Hence we call T the total expected loss statistic.
As a better indicator of the difference between the observed and the
postulated distributions we can look at the ratio of T to the
total actual cost of claims (i. e. E
f. n. ) for the sample. For a i=l 11
good fit this ratio, expressed as a percentage, should be small.
To illustrate this point, suppose that the total actual cost of
claims for a sample of data, representing a particular period of
accidents, is equal to £1,000,000. If the above defined ratio is
equal to 2%, say, then the loss we would be incurring by adopting
the hypothesized distribution under H0 as the true distribution of
claim amounts would be equal to £20,000. It would then have to
be decided, on the basis of the situation at hand, whether such a
discrepancy can be allowed. When testing the goodness-of-fit of a
predicted model to actual data, T indicates, in monetary terms, how
far from reality our model is predicting. A set of values of T
calculated from samples of actual data collected in the past and
their predicted models can indicate the reliability and consistency
of a prediction technique. This knowledge will be valuable when
setting up reserves to meet the cost of future claims as calculated
from a predicted distribution of claim amounts. I
We have given some consideration to finding the sampling distribution
of this statistic. We may use the method of derivation of the asymptotic
distribution of the Chi-square test statistic and argue as follows:
If the null hypothesis is simple, so that all the parameters of the
31
postulated distribution are known, then the probability poi that an
observation (claim amount) will fall into interval i can be calculated
on the asswnption that Ho is true. In the derivation of the
asymptotic distribution of the Chi-square statistic it is shown that
the quantities (ni-npoi)/V are approximately unit normal
variates, and the x2 distribution emerges as the sum of the squares
of these quantities (see Kendall and Stuart (1973)). If the null
hypothesis is composite so that, say, r parameters have to be
estimated from the sample, then poi will be a function of the r
unknown parameters. In the proof of the asymptotic distribution of
the X2 statistic it is again shown that when the r unknown parameters
are estimated as the solution to a set of r homogeneous linear
equations in the ni (for instance, estimation by maximum likelihood
method), then the quantities (n. -npoi)/ ö will be unit normal
variates (see Kendall and Stuart (1973)). Therefore, for the
purpose of finding the sampling distribution of T, if the above condition
in the case of estimation of the parameters holds, we can take it that
in both cases of simple and composite hypotheses the quantities
(ni-noi) where not = npoi
are normally distributed with mean zero and variance noi, i. e.
N(O, noi).
Therefore, fi(n1-noi) are independent N(O, fi2noi)
for i=1,2....... k
and .kk T=
"E1 fi(ni-noi) is distributed as N(0, E fi2noi)
=1
It is thus possible to compare the standardized value of T with the
critical values of the standard normal distribution. To justify
32
the multinormal approximation to the multinomial distribution, as in
the case of the X2 statistic, we believe that the expected frequency
in each interval should be greater than or equal to S. If this is
not so, then enough intervals will need to be pooled together to
remove this condition.
2.5 Application Procedure for the T Statistic
In this work we shall use the statistic T in addition to the
formal Chi-square goodness-of-fit test. We are interested in T as a
measure of the discrepancy, in monetary terms, between the observed
and the postulated distributions. We shall not, therefore, concern
ourselves with its sampling distribution or making comparisons of
its standardized value with tables of standard normal cumulative
percentage points. Hence intervals will not be pooled in the
calculation of T. When fitting a distribution to a sample or
comparing a predicted distribution with an observed one, the
computer programs will produce a table of results. "Expected Loss",
as defined in 2.4, will be calculated for each interval and
printed in the table. The total expected loss, T, will also be
produced. The ratio of the total expected loss to the total actual
cost of claims, calculated as mentioned earlier in 2.4, will be
shown at the bottom of the table as a percentage.
2.6 * The Kolmogorov-Smirnov Goodness-of-Fit Test
Two criticisms of the Chi-square goodness-of-fit test when used
for continuous distributions' were the necessity for grouping the
individual observations and the adoption of large intervals for
small sample sizes. Both of these procedures result in loss of
information. Besides, when there are k intervals, the 2 test
33
statistic is based on k comparisons between the observed and
expected class frequencies, while there are n observations in the
sample. Therefore, in such circumstances, it is preferable to have
available test statistics based on individual observations. Several
goodness-of-fit test statistics exist which are based on the
individual sample observations and are functions of the deviations
between the observed cumulative distribution of the sample (the
empirical distribution function) and the cumulative distribution
function under the null hypothesis. Let us first define the
empirical distribution function:
For a sample of n random observations xl, x2,....., xn we define
the empirical distribution function Sn(x) as
0 x<x(1)
Ux) = r/n x(r) ,x< X(r+1)
1 x(n) ,x
The x(r) are the order statistics of the sample. Hence Sn(x) is
simply a step function which gives the proportion of the observations
less than or equal to x.
The best known statistic of the above form is the Kolmogorov-
&nirnov test statistic. his goodness-of-fit test was first proposed
by Kolmogorov in 1933 and then developed by Smirnov in 1939. If
ro(x) is assumed to be a continuous and completely specified
population distribution function under the null hypothesis and
Sn(x) to be the step function of the sample, then the test makes use
of the statistic
34
U3 max ( Sn (x) - Fo (x) 1 x
It is expected that SnCx), for a random sample of n independent
observations, is fairly close to the specified distribution function.
If it is not close enough, then the distribution under the null
hypothesis is not the correct population distribution.
The maximum deviation D is a random variable whose sampling
distribution is lcnoum and is independent of Fo(x), when the null
hypothesis holds, provided that F0(x) is continuous (Massey (1951)).
Therefore, D is a distribution free statistic. Its limiting
distribution was derived by Kolmogorov himself. Smirnov (1948)
gave a tabulation of the limiting distribution of D. Massey
(1950-a) provided the method for evaluating tha distribution of D
for small samples. Tables for determining the significance of D
in finite samples were given by Birnbaum (1952). A table of the
critical values of the test statistic D at different significance
levels for sample sizes n=1 to 20, n= 25,30,35 and n> 35
was given by Massey (1951). For the sake of convenience, the
critical values of D for large sample sizes (n > 35) are reported,
from his paper, in table (2.1) at the end of this chapter.
When data is only available in grouped form, it is possible to
calculate the deviations ISn(xi)- Fo(xi)I at each point xi'where
xi is the upper boundary of interval i. Massey (1951) states that
grouping the observations into intervals tends to 'Lower the value of
D, and he-assert. s that for grouped data the appropriate significance
levels are smaller than those given in his table. For ]arge sa-nples,
however, grouping causes little change in the appropriate
significance levels. If the nunber of categories is small, then
important chi.. gcs can be expected in the significance levels for any ýj
sample size. According to Massey (1951), the Kolmogorov-Smirnov
Statistic, D, is correctly used only if the distribution Fo(x) is
continuous and completely specified as regards form and all its
parameters. The distribution of the maximum deviation, D, is not
known when certain parameters of the distribution have to be
estimated from the sample values. When we estimate the parameters
of the population distribution from the data, we are in effect
adjusting these parameters according to the sample values, and
in consequence we should be making a closer fit of the hypothesized
distribution to the sample values. Hence we expect that at the
same significance level the critical value of D will be smaller
than when F0(x) is completely specified. Therefore, in these
circumstances, if the maximum absolute deviation exceeds the
critical value Da(n), corresponding to a significance level a and
read from an appropriate table of the critical values of D (for
large sample sizes, n, see table (2.1)), then we cansafely
reject the null hypothesis and conclude that the population
distribution is not Fo(x).
The distribution of the Kolmogorov-Smirnov test statistic
when the parameters of Fo(x) are estimated from the sample values
depends on the form of F0(x) and is very difficult to find analytically.
Monte Carlo techniques can be used to calculate the approximate
distribution function of this test statistic for each particular
family of distributions (say, normal) Fo(x) under the null
hypothesis. Lilliefors (1967) gives a table, based on Monte Carlo
calculations, for use with the Kolmogorov-Smirnov statistic when
testing whether a set of observations is from a normal population
whose; wean and variance are not specified but must be estimated from
36
the sample. He suggests using the sample mean and variance (with
denominator n-1) as estimates of the mean and variance of the normal
population to specify F0(x). For large sample sizes (n > 40) the
critical values of D at various significance levels are
reproduced, From his paper, in table (2.2) at the end of this
chapter.
Lilliefors (1969) gives a similar table to be used when testing
whether a set of observations is from an exponential population
with unspecified mean. He suggests using the sample mean as the
mean of the exponential population.
2.7 Comparison Between the Chi-square and the Kolmogoroy-Sminiov Goodness-of-Fit Tests
Massey (1951) argues that the Kolmogorov-Smirnov test may be
always more powerful than the Chi-square test. He also points out
that the K-S test, at least at the 50 per cent power level, will
detect smaller deviations between the observed and hypothesized
distributions than will the Chi-square test. Not enough is known
about the power of either test to justify the preference for using
X2 or D for testing a completely specified hypothesis (Birnbaum (1952)).
However, Massey (1950-b) has established a lower bound to the power
of the K-S test in large samples.
We recall that two criticisms of the Chi-square test were the
grouping of the observations when individual observations were
available and the adoption of large intervals for small samples. Both
of these procedures result im.. loss of information. The Kolmogorov-
mirnov test, however, uses individual observations and hence may
utilize information more cconpletely than the Chi-square test.
For very small samples the C'hi-square test is not applicable at all
37
because its sampling distribution is not distribution free for
finite sample sizes and is not known. The K-S test, however, may
be used for very small. samrles.
The major shortcoming of the K-S test is that when the
parameters of the postulated distribution must be estimated from
the sample values the test is not applicable because the sampling
distribution of D is not distribution free and is not known. In
such circumstances the limiting distribution of the Chi-square
is easily modified by reducing the degrees of freedom.
2.8 Application Procedure For K-S Test
In our work, we can apply the Kol ogorov-Smirnov test to examine
the goodness-of-fit of a predicted distribution of claim amounts
to actual data (i. e. in situation classified under (ii) in Section
2,1). In such cases the null hypothesis is of the simple form
and we can use critical values of D given, for large n, in table
(2.1). However, in situations classified under (i) in section
2.1, when we fit a distribution with unspecified parameters to a
sample of actual data, the null hypothesis will be composite and,
as mentioned earlier, we cannot in general use the K-S goodness-of-
fit test because tables of the critical values of D do not exist
in these circumstances. There is, however, an exception in the case
of the lognormal distribution.
We say a random variable X is distributed lognormally if and only if
Y= log X is distributed normally (see Chapter 3). Y= log X is a
one-to-one function and hcnce we can use a test of normality for Y
as a test of lognormality for X. Therefore, for the lognormal
distribution we may use the Kolm. ogorov-&nirnov test statistic along
with table (2.2) of its critical values for large sample sizes. As
38
we mentioned earlier, this table has been produced by Li. lliefors (1967)
by using Monte Carlo techniques, and it is for testing the
goodness-of-fit of a normal distribution with unknown mean and
variance.
We menti. oned earlier that if parameters are estimated from the
sample values then the critical values of the K-S statistic would be
smaller than those in the standard tables (of Massey (1951), for
instance). This provides us with a means of safely rejecting a
postulated distribution when its parameters have been estimated from
the sample. For this purpose we need only to check that the absolute
maximum deviation, D, exceeds the critical value Da(n), given in
table (2.1) for large n, to conclude that the hypothesized distribution
should be rejected at significance level a. If D does not exceed
Da(n) then we cannot decide whether the null hypothesis should be
rejected or accepted.
Our accidental damage data are in grouped form. Therefore, we
explain the method of calculating the Kolmogorov-S; nirnov test
statistic, D, for this type of data. Let us assume that n
observations have been grouped into k intervals such that xi and ni
are respectively the upper boundary and the observed frequency of
interval i. Suppose that under some null hypothesis the expected
frequency for each interval has been obtained and that for interval
i it is equal to noi. The above is all the information we need to
calculate the value of the K-S test statistic, D, without resorting
to the calculation of the empirical and theoretical distribution
functions. This is because we can easily show that
D max I Sn Cx1) - Fo fxi) x. 1
E max 1 31
(l. . -1°j) I
where i=1,2,...., k.
39
Therefore, if in a table which gives the above information, a
column for (Actual-Expected) frequencies exists, we can rapidly
calculate the D test statistic. For this purpose we need to add
the successive values in this column, starting from the first
interval, and to find the maximum absolute value of the cumulative
sums which we obtain. We then divide this maximum absolute value
by n, the sample size, and the result will be the value of the
K-S test statistic D.
Because the distribution of claim amounts is skewed to the right
we expect that the largest deviations of the observed from
expected frequencies would occur in the lower tail of the
distribution. The deviations usually change sign from every
interval, or every few intervals, to the next and so it is
expected that the maximum absolute deviation, in the cumulative
sum of the (Actual-Expected) frequencies, will occur somewhere in
the laver tail of the distribution. Therefore, we need only to add
the values of (Actual-Expected) frequencies for a few lower intervals
to be able to calculate D.
If the null hypothesis is simple, then we compare the value of
D with the critical values of its distribution given in table (2.1)
for large n. If Ho is composite and the distribution under the
mill hypothesis is the two-parameter lognormal distribution, then
we use table (2.2).
Let P= Pr (D(n) a D) where D(n) is some value in the table of the
critical values of D. In other words, P is the probability of finding
a value from the distribution of the K-S statistic which is greater
than that found from our sample. According to different values of
P we can classify the result of the test as follows:
40
if (1) P>0.05
or (ii) 0.01 ,P<0.05
or (iii) P<0.01
then the difference between the observed and the postulated
distributions, under the null hypothesis, based on the given
sample is respectively:
(i) Not Significant
(ii) Almost Significant
(iii) Significant
The above arbitrary classification is based on a 5% significance
level. A different level of significance, say 101o, may be adopted
if stronger confidence is required from the test.
2.9 Other Goodness-of-Fit Tests
There are two other goodness-of-fit tests which are, like the
Kohnogorov-Sr, tirnov test, based on the deviation between the sample
empirical distribution function and the hypothesized distribution
function. These are:
(i) Cramer-Von Mises
(ii) Anderson-Darling
test statistics. For the calculation of these statistics we require
the sample order statistics, and hence we need to know the values
of the individual observations. Most data on claim amounts,
including our own AD data, are only available in grouped frequency
form which does not allow the exact calculation of these statistics.
We therefore do not consider these statistics in the present work.
41
2.10 Tables Table (2.1)
Critical values of the Kolmogorov-&nirnov Statistic Da(n),
for testing completely specified distributions. sample size =n> 35
Level of significance Ca) 0.20 0.15 0.10 0.05 0.01
D Cn, 1.07 1.14 1.22 1.36 1.63 a Vn
Table (2.2)
Critical values of the Kolmogorov-Smirnov Statistic, Da(n),
for testing a normal distribution with unspecified mean and variance . sample size =n> 40
Level of significance (a) 0.20 0.15 0.10 0.05 0.01
D (n) 0.736 0.768 0.805 0.886 1.031 a
42
CHAPTER 3
The Lognormal Distribution
3.1 Introduction
Over the past century, the lognormal distribution has emerged as
one of the most widely applied distributions in practical statistical
work. Aitchison and Brown (1957) studied this distribution thoroughly
and in 1957 published a book entitled "The Lognormal Distribution"
in which almost all the results previously found by other people are
collated. An extensive bibliography is also included. Their review
of the literature shows that this distribution has successfully fitted
data from various branches of science and engineering. A core up to
date bibliography is provided by Johnson and Kotz (1970).
References to the application of the lognormal distribution in
the field of insurance were made in section 1.4. The papers by
Benckert (1962) and Ferrara (1971) s1--em most relevant to the present
work. In both papers, the lognormal distribution is fitted to several
sets of claim size data from different branches of general insurance.
The former, however, does not define or present the data and uses only
maximum likelihood estimation formulae when individual observations
are available. In the latter reference, grouped data are available and
the three parameter lognormal distribution is fitted by a combination
of the methods of quantiles and. least squares. We shall comment
further on this procedure when we consider the estimation problem of
the three parameter case.
Aitchison and Broun (1957) deal extensively with the estimation
problem where individual observatiors are available. Their treatment
of estimation from grouped data is, however, so brief that it is
contained in a few paragraphs.
43
Most data on claim amounts, including our accidental damage data,
are avail. abl' in grouped form only. A substantial part of this chapter
is, therefore, devoted to the problem of estimation from grouped data.
The two and three parameter lognormal distributions are initially
defined and some of their properties are derived. A theoretical
justification for the emergence of the lognormal model for the
distribution of claim amounts is put forward. In section 3.6, several
tests of lognormality for the 2-parameter case are examined and the
accidental damage data isthen tested. A simulation exercise is carried
out to see how these tests perform when applied to actual samples of
lognormal data. Estimation from grouped data, for the 2-parameter
distribution, is studied in section 3.7. Several methods are
considered and, in particular, a special technique is proposed for
estimating the parameters by the method of multinomial maximum likelihood.
A computer simulation is performed to measure the efficiency of various
methods of estimation. In 3.8, the 2-parameter model is fitted to the
accidental damage data and the results are analysed. The effects of
inflation on the parameters of the model are next discussed and a
technique for predicting the distribution of claim amounts during a
future period is suggested. This is then tested on our AD data. Several
indices of prices and wages are examined to find the appropriate index
for changes in the accidental damage claim amounts over time.
The 3-parameter case is dealt with next. A graphical test of
lognormality is suggested which provides us with an approximate method
of estimating the location parameter. In section 3.12, the estimation
problem for grouped data is considered. A special technique for
estimating the parameters by the method of least squares is suggested.
The multinomiial maximum likelihood method is also modified for the 3-
parameter case. These methods are then used to fit the 3-parameter
44
ix dcl. to our accidcntal dairage data. The results are analysed iii
? 7ýý
3.13. The theoretical effects of the rate of inflation on the
parameters of the model are studied in 3.14. Predictions are made
using accidental damage data from the far past and the results are
compared with actual data from the past. Finally, the findings of
this chapter will be discussed in the conclusion section 3.15.
3.2 Definition
A random variable X is said to be lognormally distributed if 2
and only if Y= log X is normally distributed. Let u and a be
the parameters, mean and variance respectively, of the distribution of
Y.. We denote its probability density and distribution functions by
i2 fN(y; It ,? ) and N(y; .1 ,ý) respectively, : here
2 cy; 11 to
f and
"N (y; , a2 )
for -co<y<o
_ (Q / )-1 exp 2(u )2] (3.2-i)
Y fN (t; u , Q2) dt (3.2-2)
2 -Go <It <a* ; 0<a <a*
The probability density function (p. d. f. ) of X can be derived, by
using the transformation Y= log X, as
fLN(X, u , ßs2) - (Qx�Z, r)-1 exp 2( log xe- )2 (3.2-3)
2 and its distribution function, LN(x; u ,c), will be
22' LN (x; u Q) =N (log x; u a) (3.2-4)
for x> 0; -c3 <u< co ;0<Q<
2 fLN(x; p , e) , as defined in (3.2-"3), is the p. d. f. of the two-parameter . Lognormal distribution. It is evideat that u and Q2 are not location
or scale parameters for X.
45
If we rearrange (3.2-3) into
f (a, U , c'2) = LQP (x/P) /] -ý exp {- 2[ log
6x/p 2}
(3.2-5)
where p= ell
then it becomes obvious that p= ell and a are the scale and shape
parameters respectively. It will be shown later that eu is the median
of the two-parameter lognormal distribution.
We can introduce a location parameter, c, into the model by replacing
x by x-c in (3.2-3). The parameter c serves as a threshold below which
a lognormal variable X is not realized. The p. d. f. of the three-
parameter lognormal distribution is therefore
fLN(x; c, u, Q2) ._[ o(x -- c)/ ] -1 cxpf -2[ lcg(x -ac) - u12} (s. z-6)
2 for x>c; -oo<u<c; 0<a <
and its distribution function will be
22 LN(x; c, u, cr) = N(log(x - c) ;u a) (3.2-7)
In the 3-parameter case it is log(X - c) which is distributed normally
and not log X.
3.3 Properties of the Two-Parameter Lognormal Distribution
The following results can easily be derived for the two-parameter
lognormal distribution (see, for instance, Aitchison and Brown (1957)).
The distribution is unimodal and has a mode at
Xmode = exp (u - a) (3.3-1)
The median of the distribution is at
xmedi = exp(P) (3.3-2)
Moments of all orders exist and in particular the rth moment of X
about zero is
46
of = E(Xr) = exp(ru +122 ra) (3.3-3)
Heyde (1903) has shown that this distribution cannot be uniquely
determined by its moments because there exist other distributions
with the same moment sequence as {er}
The mean of the distribution, which for simplicity we denote by a
is at
a= E(X) = ei = exp(u +2 (3.3-4)
The rth central moment, er , can be expressed in terms of the rth
and lower moments of X about zero. In particular
'-e'2 (3.3-5) e2 = e2 l
3 03 = e3 - 30,101 + 2011 '; (3.3-6)
24 04 = e4 - 4e1le3 + bei e2 - 3ei (3.3-7)
2 Hence the variance of X, denoted by 6, is
,' --- 2-22
Var(X) =ß e2 = exp(2u +a )(e a - 1)
which on using (3.3-4) gives
22222 g= aA where A= ea -1 (3.3-8)
From(3.3-8) it is obvious that A is the coefficient of variation 2
of the distribution which depends on a only.
From (3.3-6) and (3.3-7), 0 and o4 can be expressed in terms of a
and a as
03 =a3 (a 6+
3), ')8
4(X12 + . 6X10 + 35x + 167
64 + 3x 04 )
Hence the coefficients of skewness ard, kurtosis of the distribution,
1 and ß2 respectively, are
47
ri=3_ A3 +3>0 (3.3-9)
ý= q4
=a+ 6x6 + 15X4 + 16x +3>3 (3.3-10) 2 ß; 7
which indicate that all lognon l densities are skewed to the right
and that t}: --y are more peaked than their related normal densities.
2 For small values of A, and hence of a, i and ß2 are close to 0
and 3 respectively. In such cases, the central portion of a lognormal
frequency curve resembles a normal curve and may be approximated by it.
From(3.3-3) it is evident that er is a product of e4 and
.22 L-9- e2 which Laurent (1963) defines as the two "functional characteristics"
of the distribution. The mean and variance of the lognormal distribution
are non-functional in the sense that two lognormal populations may
have the same mean or variance although they have different parameters
and a22. This motivates the use of the median, ell , and the
function ca as measures of central tendency and dispersion respectively.
We can show that a simple relationship exists between the quantiles
of the same order of the lognorrral and the standard normal distributions. 2
Let zq and xq be the quantiles of order q of N(z; 0,1) and LN(x; u, o )
respectively. Then by definition 2
N(zq; 0,1) =q= LN (xq; ua)
i, e. =P (X , xq) =P (log X, log xq) =P (Y , log xq) (where Y= log X)
u lo; x -u
=P (Y -
log ä =P- (Z*
Q
(where Z is the standardized normal variable)
log xQ -u =N(p, i
48
logx -u Hence za = (3.3-11)
or xq = exp (u + azq) (3.3-12)
In particular, the median of N(z; 0,1) is at z0.50 = 0, hence
from (3.3-12) the median of the distribution of X is, as mentioned
earlier, at eu .
Before concluding this section, we may note that the relative
positions of the mode, the median and the mean, i. e. 2
eu-c' , eu and eu +° respectively
prövide additional evidence that all lognormal densities are positively
skewed.
3.4 Properties of the 3-Parameter Lognormal Distribution
The transformation x}x-c to obtain the 3-parameter distribution
is only a translation along the x-axis. It leaves the shape of the
frequency curve unchanged and only shifts it by an amount c along the
x-axis.
The location measures are, therefore, increased by c and hence
Xmode =c+ exp(tt - Q) (3.4-1)
Xmedian= o+ ex' (u) (3.4-2)
i aý "mean =c+ exp(u + r) (3.4-3)
The moments about the mean and the dispersion measures remain the
same as for the 2-parameter case. In particular the variance, ßz, and
the coefficients of skewness and kurtosis are the sa: re for both the
two and three parameter lognormal distributions because they are all
functicnr, of the central rrernents. The coefficient of variation becomes
49
A' where
IM
a== (3.4-4) i. e. +a = 1+c a
which is a function of all three parameters. 2
The relationship between quantiles of order q of LN(x ; c, u, cr) and
N (z ; 0,1) will, from (3.3-12). become
X=c+ exp(11 + azq) (3.4-5)
which is equivalent to zq =Q log(xq - c) - (3.4-6)
3.5 The Lognormal Distribution as a Model for Claim Amounts
Let X0 be the amount which an insurance company at the inception
of a policy expects to pay in case of a claim on that policy (X0 can be
considered closely related to the net premium). Suppose that there
are n factors which affect the size of a claim on this policy, such
as age, type of car, district, driving skill, climate, occupation, etc.
Let Xj be the size of claim due to the effect of factors 1 to j only.
If we assume that the effect of factor j is to irodify Xj_1 to Xj by
multiplying it by a random perturbation U), i. e.
xi x_1. U
No UlU2 ... Uj
then for a policy subject to n different factors we have
X1 = Xo ujU? ... n
where . {U1, U2, ... tin } is assumed to be a sequence of random
variables With knoN joint distribution. Consequently
50
n log X1= log Xo +I log U.
j =1 -
provided that the joint distribution of U's is such that the central limit
theorem applies to the sum of their logarithms, it follows that log Il will
tend to normality for large n. Hence the amount of the claim will be
(approximately) lognormally distributed . Ferrara (1971) and Finger (1976)
also justify the lognormal nadel with arguments very similar to the above.
3.6 Tests of Lognormality :- The 2-Parameter Case
An irportant step in the identification of the statistical model is
to test if our sample is likely to be from a population whose distribution
belongs to a particular family of distributions. In other words, we should,
as a first step, test if our sample is, say, from a logformal population.
At this stage we are not interested in the parameters of the model but would
just like to know if, on the basis of the saa'nple values, the assumption of
lognornality is reasonable enough to allow further analysis according to
this model.
We consider three tests for the lognormal distribution.
3.6.1 Graphical Test
From equation (3.3-11) , i. e.
Zq .Q
lO-ý Xq Q
it is obvious that the locus of the points (log xq , zq) is a straight
lire. 1'hen these points are calculated fron a sarple and plotted on a
rectangular co-ordinate system then, if the sample is frort a t1co parameter
lognonal populaticn, the points should lie approximately on a straight
line.
To calculate zcu, quintile of order q of N(z; 0,1) distribution, ' we need
51
to know q. This can be calculated from the sample empirical
distribution function which we define as
Total number of claims <x F(x) = P( %X x) ° Total number of claims in the sample
(3 6-1)
Therefore r(x) is simply the proportion of claims with amounts less
than or equal to x. If we express this ratio as a percentage, then we
call F(x) the sample cumulative percentage function. Hence, at the point
Xq in the sample we take q= F(xq) and enter a table of the cumulative
distribution function of the standard normal distribution in order toaind
the value of the variate za. The point (log xq , zq) is thus determined.
The use of a special graph paper called the "logarithmic probability
paper" makes the above task easier. This graph paper has one of its
axes graduated logarithmically while the other is graduated according to
a standard nonral probability scale. Hence we only need to know and
plot (xq, q%). The logarithmic axis converts xa to log xq while the
probability axis converts the percentage proportion q to its corresponding
standard normal variate za.
3.6.2 The Skewness and Kurtosis Tests
As mentioned earlier, if X has a lognomal distribution then log X
is normally distributed. This suggests that a test of normality for
log X is equivalent to a test of lognoYmality for X. Therefore, we
may apply the skewness and kurtosis tests of normality on the transfonned
values, log x, and infer from its result whether the original sample
values are from a lognormal population.
This test is based on g, and g2 , the sample coefficients of skewness
and excess kurtosis respectively, for the distribution of log, X, i. e.
In- g1 1
m2
(3.6-2)
52
and m4
g2 `2-3 (3.6-3) m2
where mr is the sample central moment of order r.
Exact expressions for means and variances of gl and g2 are given in
Cramer (1946) as :
Er (gl) _0 (3.6-4)
var(g1) _ (n6( n+3
(3.6-5)
E(92) =- n6l (3.6-6)
var(g2) = 24n(n-2)(n-3)
2 (3.6-7) (n+1) (n+3) (n+5)
where n is the sample size.
we can, therefore, see by how many standard deviations g1 and g2 differ
from their mean values. If the difference is less than, for instance,
three standard deviations, then it is not significant, while if it is
greater than or equal to, say, three standard deviations, then the difference
indicates significant deviation from the assumption of normality under
which the mean and variance have been calculated.
Geary and Pearson (1938) have given the 5% and 1% probability points of
gl and (g2 + 3) for various sample sizes. In large samples, however, the
rough test of normality provided by comparing gl and g2 with the
approximate values of their standard errors, namely /
and nn respectively, would be sufficient.
3.6.3 Test in the (ýl, ßz) Plane
Let ºýý1 and 62 be the coefficients of ske-mess and kurtosis,
53
respectively, for a distribution. Barton and Dennis (1952) divide the
(ßl, 02) plane into different regions of unir, ýodal frequency curves. They
show that in the (51, ß2) plane the lognormal distribution is represented
by a straight line. This information provides us with another test of
lognormality.
Suppose that we are given several samples of independent observations
on a random variable (say, claim amounts incurred during different
and b2 , i. e. the sample periods of accident). We can calculate V -b
coefficients of skewness and kurtosis respectively. In large samples,
V -bi. and b2 should be close to their population values ß and ß2.
Therefore, we can plot the points (bl, b2), calculated for different
samples, on a rectangular co-ordinate system of axes. If the underlying
distributions of the populations from which our samples were derived are
lognormal, we would expect the points (bl, b2) to lie, approximately,
on a straight line. Therefore, with the availability of several samples,
we can test whether the underlying model for the distribution of claim
amounts is lognormal.
3.6.4 Testing the Accidental Damage Data for 2-Parameter Lognornnalit
We first applied the graphical test. Accidental Damage data for
seven different periods of accident were presented in tables (1.1) to
(1.7). For each sar, -ple, data is in grouped form and the sample cumulative
percentage function, F(xl) qi%, has been calculated at each point
x. which is the claim amount equal to the upper boundary of interval i.
Considering the range of the claim amounts, the points (xi , qi) were
plotted on a 3-cycle logarithmic probability paper which has its log
axis (for claim anx)unt) graduated for increases of x up to one thousandfold.
The plots for different periods of accident are presented in figures
(3.1-a) and (3.1-b). To avoid producing a. too voluminous th3sis, these
54
S S
Figure (3.1-a) - Points (x , F(-) ) plotted on lo, -. arithmic
probability paper .
N I'
117-17
F: T
74/iST QUARTER ;' ý'I'""'('" , ý. r. j.: l
777 f:. .tI: ,;; -ý. ̀ , fit. -:,
,i!. "', ,I:. I.., t--... '. -.
. ourl eLLLLL
arm T
'71-77-7 II ý_1fI! I mo-"j i
73/4 ýZUART
. 77
r. ...
ir
i-
t t
=i
l tS
zt
tt 1
S, -
tS
-'t
_; t
-S
i
}
'ji i
3 =
_-
__
::
:':
$it
S:
s=
_:
'_
=II
"_
-H :
"_
LY 71-
III .1 -j'r".
.. +ýI,
ý,
! ý..
", Ii''. 1 _- I ýI. I
II'
'"17ý ('...
Lý:. ý t'1 ý--
'i; I. i ý-II 1I..
-fit `ý -'i" 1 ii.
W
ý..,
i. i I
. «. ý. {..., _ýý__ý.
ý. Lýf_ý.
_L... _,. _. -t_
77
. _. , may. -tai. ..... _. ý .... ..:
ý:: "...:: ýý. ý_ ý. r ý_ -
ýf t 74/3RD QUARTER
..,
rý1+"ý;: ifli .
LL
:. j t.
ju"I: ý'.
1 1; !i
. -`-
'---`-
di
[L I
74/2ND ^L"ABT tIý.. ... ........
I " ý ý"ý '
{..:.?. ß. 1 1_:... 1. '... -. - ---ý, ý'!. L ýr . _. -ý ý iJ ý _ _ _
1 :A ß. 1 I
.+5... ..
T..... .
Fiý. ire (3. ! -b) - Poirts (x sr
(x) 'o tied on iagnrithrn! probability paper
4 T 1 T .
ýi i. «1'__l-
"'» . rl .IiII
ýý_ I
Imo,
. 'I^ . jý
.'i .4J. 1i _.
1: ý..
_ý.. i. ',:
ý';
7 -1
HL (1 ký;.. l... (.. %I:.
_li:;.. L
ýLI
44
j J
-' 75/1ST QUARTER TA
T :_ ITiý .
JI'_ 'I 111L! _l.
'T-ý"'ý'.. "`-'ý 11:.! _
... I" '.. t.. t
i
_-
s
a
V
c'
$ ,.....
...... . ý.
...... .! _..... ..... ...... _
I H
rte . 1..
FF..
...::..
1 .1
.; : ý.......:;., }. 1.:,...
?4 4TH QUARTa"Ft
LL
i
2
*
2
2
I
2
2
2
s
r ;.....
t.
ill
I
:I
F- I 4J.;
LL L
[7,
7512ND TERE L" ý-
'
IL" ,4i..
ý l. 1ý. ý_.. '. I:.: ý'ý"j j 1i. _ß ý .... _ý... l i. -+_ ; ;.....
-..
-; "ý-i
:........ ..
and other graphs have been reduced from their originals. jr, this case,
they have been reduced to 1
of their original size. However, for the
purpose of comparison between plots for different samples, it is more
convenient to present these graphs as in figures (3.1-a) and (3,1-b).
For each sample the points appear to lie approximately on a straight
line. We have fitted this line by eye. The pattern of formation of
the points about the line is the sane for all the samples. The deviations
from the line seem to be greater at the tails of the distribution.
It was deemed necessary to see what an actual sanp1e of two-
parameter lognormal observations from a population similar to those from
which our accidental damage data have been derived would look like when
plotted on the logarithmic probability paper. A computer simulation exercise
was, therefore, performed. We generated ten random samples each
consisting of 2500 lognormal observations from the population with
.2 4.5 and v=1; It will be shown later that these values of u and 2
are close to our estimates of the population parameters for the
accidental damage data. The size of each sample was adopted as 2500
because this figure is close to the size-of our AD samples.
For each simulated sample, the random lognormal variates were grouped
according to the same grouping format as for the AD samples, i. e. up to
600 in bands of 30 and afterwards in bards of 100. The computer program
P2 performed the task of simulation and then printed out a table of
grouped data with its corresponding sample cumulative percentage function.
The ten simulated samples, thus generated, were then each plotted on
logarithmic probability papers in the same way as described for the
accidental damage data. The plots are presented in figures (3.2-a),
(3.2-b) and (3.2-c).
In each case the points seem to lie approximately on the straight line
55
pure (5.2-e. ) - Fo int (. x ,) plotted on logarithmic I protabilitly paper.
(STI1ULATF. 'i) SEvý^FI3)
i- _t- 1
rk.
^'..
. ---
H i... 4 ci_ýýý. I.
'-1. ". ý_. r - ma ` "
-i
17
-Li
1" I I
ý 'Ii
" .ý 1
, L ý_ýj-"ý-
ý
LLSi . UL' w0 %VATION
7-7
i:
_x
x_ =Is
ý'-ý -_
1-7
-- . u--
ß. -.. r. 1.; '' I ..... ý.. ý.,. L.. ý_
... l.. _i-.. -1
I"
i
x
e
Y
ý;: ý... f a....... ....... ... :. i_ :ý_..:... .
::.:: .
::..... ._1
. t. I1
ii,, . 'ý I"!
i.. ,. i.:. i.; tL,... I .I,. i
.. i:. i...:
17-
-' ~i ;}; " . I. _ý...: _
'I ... -if I i.
ýTji
it.
77 7
ý,
r ;!
I.... ,. 4 '1"
. f.
i : i-Its. : L.. i::.. (:
flT r I
:i
s
s
s
s" s
sl
s sý
ss
.s
r
sý t.
DI:
r'ýTý, iýl"E: _ýT-.
ij" : -: ý i{ 'jIr: - ý` . G' _. ' - -ý = - - 1- - r-
ý 1 . 1. ý.
r' - -ý-7 -"r-. lý. r.
i
.i ý", i , 1:.. i. _: ý" 't -rýLiý_I_; ý_: ý. 1 .r ! , _. ý _... y_.. _-ýj
_ _ y
1
. 1.7. ý
.. ir. f...
'.
_r, ...., ý.
77
ý.. `.. a... ýýý"'T"ý. +.
ý i1..
r,
1... LýT_..
_;.... Y... ""ý.
2
i
0
s
e
sýý
ýJf
(3.2-^} - Points (x : `i, Y) jp1ottcd on 1crarit1=, ic
pros abil itv pr (SII4a1..
tirJ i'. ir? I
z. .. ---- t...... .
f7l
1-7
14.
77. I. ;
LL L ,Ir
{I.. r .1t ý": s.
FF1. «<f...
r1LL"i
' -I
;: IiI ii.
ý"jý. 1' I ; "" fl_T:
=1 ý1.: L"
`ý ý!
-1 . ý.
`
: i.:..; ..:
' ia..: 1. i..;
.
1.
--. I .. 1'
ýý. ...
. ýJ T ,L --T.
1
. ULL 4-L
I 1_jT;
°ý: _.
"
:. wl . 7-W
ýý :I t. f ... .'It, 1f., i,. E
i ' 1 ' -- } 1- 7 --
, i-r"-f-- ý :; i, 1ilfý! iI I`I
. ý.::
i. i i. i. 1i "? ii :I_. i.:.. ii ýtl; ý. : i:::
1Lý_J
. rr ý . I___. ý .
Y"'ý X4 1.. _: ß.,. _i: -. _
tom, _. _1 i 1! if! '" ...;.. ß. _t_1_
ýlýýý .: ýI ' I.
ý- i SIMUL ATED O
:
BS1ytVATI01
ý .
T 7,7, -177 lý
x i
z. x
=x
s"x
xx 1.
-i"c
's
-" s }
"s i
I
-. g.
:2"-
i'
s.
:Is
c =is
s= ss _. T
s
:_
.' _'
"s
`
4ý
I' I"_' i : Iý°_ : li`tý'i i ..... iii. I. i... l. t ....!
'! M1I..
I
-1. .. l'
"'l..:. ( 1.: 1:. ý..
ý' :...
ýl..:. ":
"-". I'. 1.1 .; ý_. ,_...
", --.. {
-IT id HiS, J: 1
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''1.
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. ".. L...
i tip iý;..... .. L. li .l 44
I1: t
77
: 1. ß ; Jýý'I , -{
- .; {Ii. ` j{t : ýý1: 1 !ýi j" :I
F777- -7
li-.. ý_J ____
i; '; 1
..... ..... t ... ý_
. . ý, - : ýý. ý_. ý.... -. - ---- --
gure rroi=?: L
I i7- 71 Ifi"týj I
_I' , "tiý I i: ýý-!
' t: " !ýit. _i `; 'i"t..., r..
I TT
Ifl
; "" }ýr ýt`
Sr, OL? TM 0 If, I 9"HIMTICN
2
S
F(:: ) )p CTt°d on I7i? S! ~J 1t
A! iT LE2 )
77-
r" -t'- :
ý. ý. ".
Iý ýt-} -{--; __; ý.. i,:.. ---' }... ß_1_L.
_: __1_. _::
_r. l "ýJ. ý-i I_ýiIti
IT X77:
'
s: i=
sr$ ss
:
s
I .:
r
I
" i.
E
. ýýý". .,. . ur"" .. " .t f
F
" 'd
"t
which has been fitted by eye. In the lower tail of the distribution
the points seem to lie almost exactly on the line , unlike the behaviour
of the accidental damage samples. The pattern of the points in the
upper tail of the simulated samples is similar to that of the accidental
damage data, i. e. the deviations from the line are more marked. This
indicates that even from a sample of 2500 lognormal observations we
cannot obtain a close fit to the straight line in the upper tail of the
distribution. This is because the lognormal is a very skew distribution
and observations in the tail are scarce. Therefore, we need a much
larger sample size before we can see a true picture of the upper tail of
the distribution. Hence we can safely assume that the pattern of the
points in the upper tail of the accidental damage samples is by no means
unusual to the lognormal distribution and should, therefore, feel
satisfied that the distribution fits the upper tail reasonably well.
The simulated samples, however, tell a different story in the. lower
tail of the distribution. Here the points lie very much closely on
the straight line-and the deviations from it are much smaller than in
the case of the accidental damage samples. Therefore, the pattern of
the points in the lower tail of the accidental damage samples is not
consistent with that of the two-parameter lognormal distribution. We
believe that this inconsistency could be partly because of the in-
sufficiencies and inaccuracies of the data in the lower tail. Most
policy holders do not claim for small amounts for fear of losing their
full entitlement to no claim discount which, in some cases, is worth
more than the amount they may recover by a claim. Therefore, the data
in the lower intervals may be incomplete. On the whole, the assumption
of two-parameter lognorriality as tested by this method seems reasonable
enough to allow further analysis.
Two other tests of lognormality, mentioned in 3.6.2, are the
skewness and kurtosis tests of normality. The values of the coefficients
56
gl and g2 for each saq)le of the accidental damage data were calculated
by program Pl. These were presented in tables (1.1) to (1.7) and are
produced in table (3.1) at the end of this chapter. The standard
errors of these coefficients, for large samples, have been calculated
according to the formulae of section 3.6.2. These are presented in
table (3.1. ) as well. From this table it is apparent that in four periods,
namely 74/3rd quarter to 75/2nd quarter, the value of gl is greater than
three times its standard error. Therefore, in these four cases, the
underlying assumption of normality. for log X, and hence of two-parameter
lognormality for X (the claim amount) should be rejected. For the
remaining three periods, namely 73/4th quarter to 74/2nd quarter, the
lognormal assumption for the distribution of claim amounts is supported.
The values of g2 are significantly larger than their standard errors.
Therefore, this test suggests that the assumption of two-parameter log-
normality should be rejected. gl and g2 were calculated from grouped
data with the assumption made, in section 1.6, about the concentration
of the claims in each band at the mid-point of that band. This could
affect the values of gl and g2 calculated from grouped data. A
Sheppard's correction for the calculation of moments of log Xis not
applicable exactly because the intervals of log X are not of equal
length.
It was decided to put the ten simulated lognormal samples to these
tests. The values of g1 and g2 were calculated from the simulated
grouped data in exactly the sane way as for the accidental damage data.
Their standard errors for a sample of size 2500 were also calculated.
The results are presented in table (3.2). The values of gl compare
favourably with its standard error and, therefore, this test performs
very well. The values of g2 are in most cases acceptable, but in the
case of samples 1,7 and 8, E2 is greater than three times its standard
error.
57
Therefore, the coefficients g1 and g2, even when calculated from grouped
data, can be used to test normality for log X and, on the basis of the
simulated samples, it seems that gl can indicate normality more
accurately than g2. The comparison of the values of gl and g2 calculated
from accidental damage samples with those for the simulated samples shows
that the assumption of two-parameter lognonnality is, on the whole,
reasonable.
Another test of lognormality is the test in (ßl, ß2) plane
which was mentioned in section 3.4.3. The values of ºý1 and b2 for
the AD samples were calculated by program P1 and they were shown in
tables (1.1) to (1.7). Figure (3.3) shows a plot of the points (bl, b2).
These points, although some of them are very near each other, seem to
lie approximately on a straight line. This supports the assumption of
lognormality as the underlying model for the distribution of claim
amounts.
To see how this test performs when applied to actual lognormal
samples, bl and b2 were calculated for each of the ten simulated samples.
The points (bl, b2) are plotted in figure (3.4). They appear to lie
approximately on a straight line. Therefore, this test can be applied to
test lognormality. However, for reliable results we need a large
number of points and hence a large number of samples.
58
ýiý
}GC
if
9a
70
6c
IC
6, ip 20 -"�0 A. u -0
Figure (3.4) - Plot of the (bl , ib2) Points for the o; TMulated samples.
3.7 Estimation of the Parameters of the 2-Parameter Lognormal
Distribution
Aitchison C-1-id Brown (1957) investigated thoroughly the problem of
estimating tho parameters of the lognormal distribution when the value
of each individual observation in the sample is available. Ibwever,
their treatment of the problem in the case of grouped data is very brief
and they point out that there are certain difficulties in the various
methods of estimation. In this section we study the problem of
estimation from grouped data. Some of the methods, described below,
have been modified from methods of estimation for when individual sample
values are given. Some others, such as the graphical, the least
squares, and the multinomial maximum likelihood methods are directly
suitable for grouped data.
Let us assume that we have a sample of grouped data where n independent
random observations on a random variable X (in our case, the claim
amount) have been grouped according to their size into k intervals.
Further, let ni be the number of observations (clams) in the class
interval (xi_1, xi) such that
k n=ýn;
1y i=1
It is usual, in the case of grouped data, to assume that all the values
within any interval are concentrated at the mean of the portion of the
distribution over that interval. If the distribution is the correct
rodcl for the population from which the sample values have been derived,
then for every relatively small interval, the mean of the portion of the
distribution over that interiral can be considered as equal to the mean
of sample values in the sp-me interval. Information on the average
amount of claim in any interval was provided by the insurance company.
This showed that for every interval the average amount was very nearly
59
equal to the value of the idd-point of that interval. We, therefore,
feel justified in asstrdng that within any interval (xi_l, xi) the
claims are concentrated at the point
iL (xi-1 * Yid
3.7.1 The Method of Moments
(3.7-1)
Let Sl and S2 be respectively the first and the second moments
of the sample values abcut zero, i. e.
1k S1 =n lýl
nixi (3.7-2)
k2 S2 nil nixi (3.7-3)
Estimation by the method of mments consists of putting S1 and S2 equal
to their corresponding theoretical values from the distribution. Hence,
by using equation (3.3-3) with r=1 and r=2 respectively, we will
have
S1 = exp (u +2 Q2 )
2 S2 = exp (2p + 2Q )
2 Solving the above sL'Tultareous equations for u: nd a we obtain
2S2 log S2
and
Q2 - log S2 -2 log Sl
(3.7-4)
(3.7-5)
(3.7-6)
(3.7-7)
^z 2 where u and Q are the estimates of p and a respectively. We have
given some consideration to the question of applying a Sheppard's
correction to the second sample moment. however, numerical investigations
60
showed that this correction does not have a significant impact on the
values of u and e2 . It has been suggested by some authors (see for
instance, Geary and Pearson (1938)) that unless the grouping interval
is more than one-third of the standard deviation of the sample, it is not
really necessary to apply any corrections to the moments. A glance at
tables (1.1) to (1.7) shows that for our AD samples the interval width
for the majority of the bands, containing some 9510 of the total number
of claims, is 30, while the standard deviation in each sample is at
least six times that. From the above considerations a correction to the
second moment is not deemed necessary.
3.7.2 The Approximate Maximum Likelihood Method
The word "approximate" in the title of this method is used because
of the assumption of concentration of the claims in every interval at
the mid-point of that interval. For a grouped sample, using the notation
already adopted, the likelihood function L is
2-kk -n1 " L= (2, ra)n/2 I1 xi exp 'ý' I ni (log xi -u )2 (3.7-8)
i. =1 2a i=l
I
2 The same values of u and a maximize both L and log L where
log L=-n log 27r -2 log Q2 k
nlog x-1 i=1 ii
2a
kz ill (log xi-u)
i=1
(3.7-9)
2 The values of u and a which r, mxlmize log L are the solutions of the follow-
ing two simultaneous equations,
u1°ß _0
loh I, =p 2
t: ence, it can be easily shoum that p and Q the estimates of u and 2
61
2 a respectively are
k
iý =nG 1=1
and
k ^ý_ 1 Cl` n i=1
n1 log xi
2 ^2 nl(log xi) -u (3.7-11)
3.7.3 The Method of Quantiles
The relationship (3.3-12) between the same order quantiles of the
lognormal and the standard normal distributions, namely
xq = eu+ a zq
is the basis of this method. Let xql and a2 be two different quantiles
of orders al and q2 caleulatcd from the sample. These can be either
selected as the upper boundaries of two different intervals or be
calculated by interpolation in the intervals of X. The corresponding
standard normal quantiles, zql and zq2 respectively, can be calculated
by reference to a table of N(0,1) distribution function. and a2 the 2
estimates of u and a respectively, are the solutions of the following
simultaneous equations
x= eu+ czai aý
A x eul azg2 q2
Hence z log x-z log x ^ _.
3c g' ch (3.7-12) zq2 - LQ1
and
ä2 log xa, log xq1 `2
Zq2 Zql
62
It can be shown that this method is most efficient when the two quantiles
are symmetrically placed (Aitchison and Brown (1957)). Let, therefore
ql =q and q2 =1-q where q< 1/2
then
zal =zq=- t, say, where t>0since q<1/2
and z2 = '1_a =
Hence u and Q2 will be
u=2( log xl_q + log xq )
and
(3.7-14)
ý2 4t2
(log xl_q - log xq) 2 (3.7-15)
Aitchison and Brown (1957) state that quantiles of order 27% and 73%
estimate u with 81% efficiency and quantiles of order 7% and 93% estimate 2
c with an efficiency of 65%. We will later use these two sets of 2
quantiles to estimate u and a from the AD samples. To obtain the values
of the c{uantiles linear interpolation in the intervals of x will be used.
3.7.4 The Method of Median and Coefficient of Variation
This is a very simple method which involves calculating the sample
median and coefficient of variation, xmedian and V respectively, and
putting them equal to their theoretical values fron the distribution.
To find xmedian' in the intervals of x may be necessary.
For the sanple coefficient of variation, V, we need to calculate the
sample mean and standard deviation. The expressions for the median and
coefficient of variation for the two-parameter lognormal distribution
were given by (3.3-2) and (3.3-8) respectively. Hence
inedian = ell
63
and 2 V_= (e - 1) 2
2 The estimates of i and a will therefore be
}log median (3.7-16)
and a2= log (1 + v2) (3.7-17 )
This method can be considered as a combination of the methods of moments
and quantiles.
3.7.5 The Graphical Method
This method is derived as a follow-up of the graphical test of
loiormality mentioned earlier in section 3.6.1. Therefore, we know that
the locus of the points (log xq, zq) is the straight line
Z. =1 log x- 11
Let us assume. that we have followed the procedure of section 3.6.1 and
plotted the points (log xq , zq) from a sample of grouped data. Let us
further assume that these points seem to lie approximately on a straight
line. We can fit this line to the points by eye. The inverse of the
gradient of the line will give an estimate of a and then from the intercept
of the line we find the estimate of p.
Aitchison and Brown (1957) suggest the following method when the
logarithmic probability paper has been used to plot the points :
From a table of the standard normal distribution function we notice that
116% Z158ö -1
and =_+1 284% 284.13%
0 Z5oä =
64
Therefore, if we put the logno: 7na1 population quantiles of orders 16 a,
50% and 84% equal to their corresponding values read from the straight
line we will have
u-Q x16%% =e
u X50$ =e
u+Q X84% =e
Hence from (3.5-18)
a u= log X50%
Then from (3.5-19)
ea _ X50%
x160
and from (3.5-20)
Q X84% e= X50%
a better estimate of eo would be
ee X50%
+ X84%
2. X16% X50%
Hence
(3.7-1ß)
(3.7-3.9)
(3.7-20)
(3.7-21)
X50*- 112 Z_ log 2(a+ X84% ) (3.7-22)
X16$ 50$
For grouped data, this method has the advantage that the data is already
in the form required by it.
3.7.6 The Method of Least Squares
This is a variation of the graphical method. Instead of fitting a
straight line by eye to the set of points (log xqj zq) we use the least
65
squares regression method to determine its equation. The theory of
linear regression tells us that if the ur_creighted least squares li:: c
through a set of m points (Ui, Vi), where i=1,2, ... , in , is
V= aU+b
then the coefficients a and b are
_W- uv
aÜÜ2
and b. = V- aU
mm where U= 1Iu V= 1 V.
M i=1 1m i=1 1
(3.7-23)
(3.7-24)
(3.7-25)
M2m
U=1 Ui and UV =1 UiV. m i_l m i=l i
As we mentioned earlier, the point (log x, z) lies on the straight
line
z=Q log x-Q (3.7-26)
Therefore, we can let
U. = log xal and Vi = zq. i
where xqi is the upper boundary of interval i, qi is the value of the
sample cumulative percentage function at the point äi and zqi is
the standard normal variate corresponding to qi, i. e.
Z qi 2
q1 exp 2) dt
The problem then is to calculate the coefficients a and b of the line
V= aU+b
66
If our grouped data consists of k intervals, then i takes integer
values frone 1 to k-1 because for the final interval ak =1 which
results in an indeterminate value of zq. The values of zq can be
obtained from a table of the standard normal cumulative distribution
function. If this estimation procedure is programmed on the computer, an
approximate formula for calculating zq from q should be used. Hastings
(1968) gives the following approximate fornula for calculating the
standard normal variate ,z, from ,p, the area under the upper tail
of its - frequency curve :
where
- 2.515517 + 0.802853t + 0.010328r
Z=t 1+1.432788t + 0.1ß9269r + 13 8rt (3.7-27
p_ 27r)-1
r. = log 2
P
J"0 exp (- 22)de
z
and t=
, 0<p<0.5
To use this formula in conjunction with the estimation method, we have
to take p=1- qi where qi is as defined above.
Let, therefore, V= aU +b be the equation of the least squares line
where a and b have been calculated according to the formulae (3.7-24) and
(3.7-25). A comparison between the equation of this line and that of
its equivalence given by (3.7-26) shows that
a= l/Q and b- --u/a
a Hence u=- b/a
and a2 _ a-2
(3.7-28)
0 (3.7-29)
3.7.7 The Mhultinomial Maximum Likelihood method
This is by far the nxost suitable method for estimation of the
67
parameters from grouped data. Let us asswre that a sample of data as
defined in section 3.7 is available. If we denote by pi the probabilit; "
that the amount of a claim will fall in the interval (xi_1 - xi), then
22 pi = LN(x1; u, Q )' LN(Xi-1; u, a )
"_ -u pi=N(a ; 0,1)-N (
Y'
Therefore, L, the likelihood function of the sample will be
(3.7-3i)
n: n. L= Pi i (3.7-32)
n i=1
i=1 1
and that part of the loglikelihood function which depends on the parameters 2
V and will be
k yi- u yi-1- u 'j log L== nl log
[N( 0,1) - N(
Q; 0,1) I
J i-1 (3.7-33)
2 The multinomial raximun likelihood estimates of u and a are obtained by
2 maximising L (or log L) simultaneously with respect to µ and a
2 Therefore, u and a are the solutions of the simultaneous equations
alogL =0
and alogL =0 a02
From these equations we cannot find an analytic solution for u and Q2
Gjeddebaek (1949) has considered this problem and introduces and tabulates
No functions to facilitate the interpolation involved in the solution of
the above equations. His mrýethod, however, becomes very laborious even
(3.7-30)
; 0,1 ) where y- lag x
68
when the number of intervals is greater than 3. It is possible to
use an iterative procedure to solve the system of non-linear equations 2
in u and a. Tallis and Young (1962) apply this method in the case of
the 3-parameter lognormal distribution.
We instead propose the use of an iterative method which maximizes 2
log L itself with respect to the parameters u and a. There are now
many computer routines-available for maximizing a general function of 1
severable variables. We use one such routine provided by the NAG library
of programs. This routine minimizes the function - log L (i. e. it
maximizes log L) with respect to the unlaiown parameters by using a revised
quasi-Newton method implemented by Gill and Murray (1972). Analytical
derivatives of the function need not be supplied. To start the iterative
process the initial values for u and a- are obtained by one of the
simpler estimation methods described earlier. The procedure uses
estimates of the gradient and curvature of the objective function to
generate a sequence of points which are intended to converge to a 4 42
local maximum of log L. V and a thus obtained can then be used as the
starting point in order to make sure that they are indeed the optimum
points. If on inspecting a maximum found by the routine it is suspected
that a 'better' solution exists, the remedy is to re-run the routine
from another starting point closer to the expected solution. With the
wide availability of the computer this is very much easier than the
methods of Gjedd. ebaek or Tallis-and Young.
In conclusion, we point out that there is another suitable method
of estimation from grouped data which is called 'the modified Chi-square
minimum method'. This involves minimizing a modified Chi-square type
statistic with respect to the unknown parameters. Cramer (1946).
1- Nottingham Algoritiunic Group
69
however, shows that this method is identical with the iruitinomial
naximtn likelihood method. Therefore, we shall not consider the
modified Cri-square minimum meted in this work.
3.7.8 Measuring the Performance of Various Methods of Estimation
A simulation exercise was carried out to see how well different
methods of estimation perform in practice. 100 samples, each of size
2500, were generated from the lognormal population with parameters 2
u=4.5 and a=I. These values for the parameters will be shown to
be close to the values for our accidental damage (AD) data. The size
of the simulated samples, 2500 , is typical of the size of our accidental
damage samples. In this way, simulated samples resembling the actual
AD samples were produced. The computer program P3 was written for this
task and samples of data grouped in the same format of intervals as for
the accidental damage data were created.
We define the numerical measures of efficiency of a particular method of 2
estimating u and a as
N Eff(p) =R CUl u)2 (3.7-34)
i=1
Eff (Q2) N0-a )2 (3.7-35) i=1
where u=4.5 $ a2 =1 and N= 100, and a1 are the estimates of
and a2respectively, arrived at by the method in question from
sample i.. For each pair of estimates (p ,Q we can find, from
forrulae (3.3-4) and (3.3-8)9 ai and $i which are the estimates of
a and a the mean and standard deviation of the lognormal distribution
respectively. ., i and ßi are then considered as estimates obtained
70
by the same method of esti. ̂ mation which was used to find ui and cri from
sample i. Because a and 0 are functions-of both u and a, it is 2
important to kiow how well a particular , method of estimation performs in estimating a and ß. 't'herefore, we can similarly define measures of
efficiency for a and ß as
N 12 Eff(a) =N1 (ai - a) (3.7-36)
i=1
N Eff(ß) - (ßi - ß)2 (3.7-37)
i=1
2 where, from 4.5 and a=1 we have, a= 148.41 and 8= 194.54.
Men comparing two different methods of estimating a certain parameter,
the one whose calculated measure of efficiency is smaller will be fudged
as more efficient and hence will be preferred. 2
The computer program P4 was written to estimate u and a, from
each of the 100 simulated samples, by the Methods of moments, approximate
maximum likelihood, 27 and 73% quantiles, 7 and 93% quantiles, median and
coefficient of variation and the least squares regression. The graphical.
method was excluded from the exercise because the task of plotting 100
sets of data points would have been very laborious. Besides, 'the
efficiency of this method depends on how experienced the analyst is in
using the graph paper and fitting the line by eye. The multinomial
maximum likelihood method was not included because it is the most
suitable method for estimating the parameters from grouped data and is,
therefore, sure to be very efficient. 2
Program P4 produced estimates of p, v, a and e, by various methods,
from each sample and calculated the efficiency of each method in
estimating each parameter. Table (3.3) Shows the results of this
simulation exercise. It is apparent that amongst the methods considered
71
the approximate it . xirumi Likelihood is the Timst efficient in estimating
the parameters from the particular type of data we are dealing, with. The
method of ?7 and 73% quantiles is the second best and nits use should be
recommended when simplicity in calculations rather than maximum accuracy
is the criterion. The efficiency of the method of median and coefficient
of variation for estimating u is the same as that of the method of 27 and 2
73% quantiles, and for estimating e is the same as that of the method Ilk
of moments. Therefore, the median can be used to obtain a good estimate
of v. The method of 7 and 93% quantiles is not very accurate for our
type of data. The methods of it inents and least squares regression
-" appear to perform with nearly the same degree of accuracy.
3.8-Application of the 2-Parameter Lognormal Model to the Accidental
Damage Data
The results of the tests of lognorrality carried out in section 3.6.4
seemed reasonable enough to encourage us to fit the two-parameter log-
normal model. to the accidental damage data presented in tables (1.1) to
(1.7). An extensive computer program, PS , was written to estimate the 2
parameters i and a by all the methods described in section 3.7, except
the methods of median and coefficient of variation and multinoi.. ia1
maxiim. m likelihood. The former method was excluded because it can be
considered as a combination of the methods of moments and quantiles.
The latter will be dealt with separately. For each sample of data
supplied as input, the program provides six relevant tables of results.
For a particular period of accident, nsitily the 4th quarter of 1973
which was presented in table (1.1), the results of a run of program P5
are presented in tables (3.4-a) to (3.4-f). The first table, (3.4 -s),
presents the estimates of u and a by different methods as well as some
of their corresponding relevant distribution statistics. Table (3.4-b)
72
gives the expected number of claims, in each interval, calculated by
using various sets of estimates. A table of actual minus expected.
frequencies is produced as table (3.4-c). To allow the calculation or
the'Chi-square statistic, values of (A
E E) for each interval up
to ? 000 and each set of parameters are presented in table (3.4-d).
It was considered that for claim arounts greater than X1000 the expected
number of claims in each interval would be less than 5. The total i
of the (AEE
values for each method of estimation and for intervals
up to £1000 is provided, but these-totals need to be adjusted for
intervals above £1000. To allow comparison between the actual cost of
claims and that expected under the model by different methods of
estimation, table (3.4-e) is produced. Each entry in this table is the
product of the number of claims in an interval, and the average anount
of claim in the sane interval (tr e average is assumed to be equal to te
mid-point of the interval). Table (3.4-f) provides the expected loss
and the T statistic, as defined in section 2.4 of chapter 2.
Program P5 was run with data from other periods of accident as well. To
avoid inclusion of a large number of tables, only the first table of
each run, similar to table (3.4-a) are presented here in tables (3.5) 'Co
(3.10). Relevant statistics such as the Chi-square, the T and the ratio
of T to total actual cost have been added to each of these tables.
Let us now closely examine the results produced for 73/4th quarter.
A glance at table (3.4-a) shows that with the exception of the method of
7 and 93% quantiles all the methods have produced almost similar results.
Estimates of u are about 4.5 and of a2 about 0.9. The skewness and
kurtosis of the frequency curve of the model are greater than those of the
sample (see table (1.1)), hence indicating that the frequency curve of
the model has a ]onger tail and is more peaked than the histogram of the
sample values. Table (3.4-c) indicates that the disagreement between
73
the actual and expected number of claims is more marked in a few of the
lo: -: er tail intervals. These differences are smallest for the approximate
maximrai likelihood ; method. Table (3.4-d) in fact shows that the Chi-
square statistic is smallest for that method although all the values of
this statistic indicate a significant difference between the sample
values and the model. It can be observed that the major contribution
to the Chi-square statistic is from one or two cells in the lower tail
of the distribution. We may recall that when we tested the data for
lognormality, the points in the lower tail did not fit the straight
line very well. We blamed the insufficiencies of the data in the lower
tail intervals for this behaviour. The fit of the model to the data
would have been very satisfactory if it were not for these one or two
large contributions. In chapter 7, where we investigate the truncated
lognormal distribution, the effects of discarding various intervals in
the lower tail will be studied.
Table (3.4-e) indicates that despite the large Chi-square values, the
total cost of claims from the sample and the model are fairly close to
each other. It is also inferred that although in terms of the number
of claims only about 4% of the total lies in the upper tail, (greater
than £600), in terms of the cost of claims about 17% of the total comes
from the upper tail. Table (3.4-f) shows that the total expected loss
is not, in fact, very large, except for 7 and 93% quantiles method and
that the ratio of T to the total actual cost is quite small.
For the method of approximate maximum likelihood we can apply the
Kolmogorev-Smirnov test of goodness-of-fit in conjunction with table
(2.2). The value of this statistic is D=0.023 which gives P <0"01 (see
section 2.8) . Therefore this test, like the Chi-square, indicates
significant differences between the model and the actual Sanple.
74
I
For other periods of accident results similar to those of the 73/4th
quarter can be observed fron tables (3.5) to (3.1o).
The above analysis supports the findings of the simulation exercise of
section 3.7.8. Pinongst the methods considered, the approximate
maximum likelihood gives the best fit of the model to the actual data
in terms of the goodness-of-fit tests. The fact that for each sample,
the Chi-square statistic is smallest for this method is not surprising
if we recall the remarks of chapter 2 about the estimation method
and its effects on the Chi-square statistic. The method of 27 and
73% quantiles also produces satisfactory results and its use is recommended
when a quick solution is required.
Before commenting on the-suitability of this model for our type of data
we will consider the estimation of the parameters by the method of
multinomi. al maximum likelihood.
Program P6 estimates the parameters by the technique described in
section 3.7.7 and provides a table of results. For our accidental
damage data, the results are presented in tables (3.11) to (3.17). For
each sample, estimates of u and a2, and the values of the mean and
standard deviation of the claim amount are provided. The tables are self-
explanatory. If, for each sample, we compare the results of this method
with those of the approximate maxiiraua likelihood, we will notice that
the estimates are very close to each other but a slightly better fit
has been provided by the present method. The Chi-square and the total
expected loss statistics are smaller for each sample. The values of the
Chi-square statistic still indicate significant differences between
the model and the sample values. This can be seen to be supported by
the Kolmogorov-Smirnov statistic. The actual minus expected frequencies
are again large in just a few of the intervals and the pattern appears
75
to be consistent for all the samples. It is, however, encouraging that
the values of the test statistics are not drastically large but
that they are just about significant.
The analysis so far shows that the method of multinomial maximum
likelihood, as we expected, produces more satisfactory results when
estimating the parameters from grouped data. Therefore, guided by this
information, it will be the principal method which we shall use throughout
the forthcoming stages of the present work.
To see how closely the frequency curve of the model approximates
the histogram of the sample values, computer program P7 was written to
plot both of them. The graphs for various samples are presented in
figures (3.5.1) to (3.5.7).
(This section is continued after Figure (3.5.7)).
76
t{ ý'.. tiaýýr C'J
73.
77
1,.
1f;:
17.
16
13.
12.
11.
13.
S.
7:
61
31 2
6
l4-1 '. i
: 2.
ý.
h'' . tr..
Eigare (3.5.1)
hI , TC: rPAM or 73. '1111 GUA1 Tf. F L TA P. ND ""PAf.
L(ý;, Nhf`': AL P. D. r. W'il1, MULTN. NAXL. f'AP4M! 11 ..
MCwýt. SIE. SICMA
2 20 -i(l, ; 270 2eO -ý3, " 260 1 2t, 1Ü 1C S1O 51C 370 (>CC 1
OF 73%1TF+ GUARTE. R DATA AND 2-PAP.:
F. C. E". WITH FILLTN. MAXL. PAP. AiE. TCks
SIGMA SL'. =i.:: 55
iýýIj: i{S`ý1: : ý: ýi '. '% . I1ý~
ýý ýý (3-5-2)
IFr.?. ai_4k '( 4
2I.
2o 'l
s.
1 f; =7. IFS
IJ
13.
12
3 22 Gf1 ; 2u afi ; CC 213 ?T.. 7C. cC 2.2 2aG 9c0 15e 9c;.. 51
mac,? _, ': cýý
Z.
f, (X
CUAPTr. P C; TA AND 7-PAP.
ri tU. LTN .I A)(L . PAP. API': TC. R .
SIGNA SQ. -. a . 257
-': rr
NI+TC': PAN or "1'IL', T (JUMr1-I' CATA fNL1 : -PW.
LOC"NOR! '. AL f . CO. r . Gl tti i: uLTN. I! AXL . PAI'"hll! 11'. K4,
or, (ý f; ;2 fl, f, : ". tý :4 I'l, :
: ':, Figure (3-5-311,
22 HISIC: F', AI1 Or- 7"1/2N("I :; t1AF*ICR L' TA AND . -PAS:.
iý. L03NOF. MAL P. i. r. '.! Illy ilU11N. MXt_. r. AF'ßtVýTFRt; 1'
tii i MC+wý1 . '" i6 SIC"T7r 5O. -l . "';?
1(. I . 7.
1 rl'
1S:
lw
13.
12.
11.
C',
E:
5. .
3.
2:
6ý g0,12C X56 T£; 3 213 24.0 27C :? ZO 3< 21, C _3 120 13 18,1 5<., 1C 57< Ef'E
1? ý1 ý. E'i
Z.
ý,
G4; ß
'TER DATA AND 2-PPM..
ILTN. f1AXL. PF. r Ar'-JrFG
SIGMA SO. -i.:: 3
-z : ci OX4 -1 L:
F. ure (3.5.4)
!t I "'. ' ]V+ ýý f
. s.
F; I , tt1 . ('\ft ; 'F 'i 1' :. GAf '[$ EINTA ANU '-. i'AP. 1
ßt7 LOL"NC''`tt. f'W1 1h ULI N . h! AXt.,
19 r1c_y_q. fý? ' :, Trt1A ýyL, "'=1 .ý 11
1C. 17.
16
In
13.
11.
9
3. 2.
ý 50 12 ý. P . 6; 'I 212 24[a 270 210(1 2: T 35 ":: 7 1 F' 20 1'" 1c 513 59[: 5? i b 3f Co
x1a-1 5. ee
1 . 5C
i. Cp
2. ßfr.
1.
G(11C;
HISTCLR; 41 Cr 79/3; r, ý-LAP. TC. R DATA AND 2-PAP.
LO; NCr"AL P. C. r. WITH MULTN. MAXL . PA. RAM': TFRS:
MEW-1. G37 SIGMA SO . -1.. ^.. '1
I -
. '
.; e ; 2;
Pip. -are (3.5.5)
'ArIF fi icy
?i 51. ̂ . 51C 5-0 F P.:
Xlü-ý 5. ce
Frr-±uvnc:
9. Se.
2.: C
...
STCC-PAM Or 7 . '1TFi GUART1 DATA AND 1-i AP. '.
GNQF.! 1AL P. C F. WITH (1ULTN. f+AXL. PAPPMý: TFPý;
W--. 571 SIGMA ýc. -1.0,55
2 13 ff .
I. ' _. *---"-" ---.. --. -...
r
7,
ii
1i
1
i
1
1
Figure (3.5.6)
A ; Mr
X13-t 5 .e
FreuC
9
4 . cO-
0
Qt:
{iIGTO. PAM ^` :5 1ST CIAP. Tf_P. DATA AND 'l. -PAP..
LO NOFMAL P. C: P. WITH i-, ULTN. MA)(L. PAPAMETC. P:,
MLWa J. 6cý; 4 SIGMA SGG. --1. C^}
- ; :0 : i. IT
2
2i
ti
ti
p
i
I: t'.
t
Fibure (3.5.7)
:(
x, a-, 5M,
9 . 'ý
9. (;
7 r.
7"C
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1.?
CATA AND 2-PAR.
MAXI.. PA. PAMC-TFF. G
SIGMA SO. -I. 2ý6
: "; c : yt' c: """ �". " i;
(section 3.8 continued)
The multinomial maxinnm likelihood estimates have been used to plot
the frequency curve. To -represent clearly the behaviour of the tail of
the distribution, each curve consists of two parts. The graph at the
top of the page represents the distribution up to £600 claim amounts,
i. e. the lower tail. The bottom graph shows the upper tail of the
distribution, i. e. claim amounts greater than £600.
As previous results indicated, the discrepancy between the frequency
curve and the histogram for each sample is more marked at the lower
tail of the distribution. The possible reasons for this were mentioned
earlier. The curve appears to model the histogram satisfactorily at
the middle part and the upper tail of the distribution. The distinct
mode and the long tail of the histogram are both portrayed by the frequency
curve. The graphs show that shifting the curve, slightly, along the x
axis (i. e. a 3-parameter lognormal model) or ignoring the lower tail
(i. e. a truncated distribution) would produce models which would fit
the actual data more closely.
Judging by the results of the foregoing analysis, we are not
satisfied that the two-parameter lognormal distribution is the correct
model for the distribution of accidental damage claim amounts. However,
the analysis has been constructive in showing how the model should be
applied in practice. It has also been shown that, for our data, a
modification of the model would provide better results. To complete the
work on the two-parameter lognormal distribution we will study, in the
next section, the effect of inflation on the parameters of the model
and will suggest a means of-predicting the model of the future
distribution of claim amounts.
77
3.9 Prediction of the Claim Amount Distribution :- The 2-Parameter
Lognormal Model
We are interested in the cost of future claims for both the
calculation of premium rates and the setting up of reserves. -lie
cannot, therefore, wait until all the claims have been notified and
settled in order to find the cost of the claims. One way of solving
this problem is by predicting the cost of the future claims. In this
section we will suggest a technique by which the distribution of the
claim amounts during any future period can be predicted. From this
distribution, and assuming that we have a knowledge of the number of
claims in the particular period , all the required information about the
cost of claims can be obtained. The technique will be tested by using
past data. The performance of this method will be examined by goodness-
of-fit tests. In particular, the total expected loss statistic, T,
will indicate, in monetary terns, how accurately the technique is
performing in predicting the total cost of future claims.
3.9.1 The Effects of Inflation on the Parameters ' of' the' Mädel
In practice, it is usually observed that the same kind of claim
costs more to settle than it previously did in the past. From tables
(1.1) to (1.7) we can see that the mean claim amount has increased from
one quarter to the next. It is generally believed that inflation is
the main cause of the increase in the cost of claims over time. The
increase may be due to other apparent factors the real roots of which lie
in inflation. For instance, in the third party liability claims, the
decision of the courts in awarding higher Compensations stems from the
rise in the cost of living and higher wages.
Let us assume that the effect of inflation is to increase a claim of
size X to X(1 + i) after a period of time where i is the effective rate
78
of inflation during that period. If we assume that the distribution
of X is the two-parameter lognormal with parameters 2 (u, a ), by a
transformation of variables it can be shown that X(1 + i) is distributed 2
as the two-parameter lognormal with parameters (p +d, a ) where we call
d= log(1 + i) the force of inflation. Therefore, inflation affects 2
only p and leaves a unchanged. If we look at the multinomial maxirum
likelihood estimates ofp , given in tables (3.11) to (3.17), we notice
that they have generally increased from one period to the next. The
mean and the standard deviation of the lognormal distribution, a and 2
respectively, are functions of both p and a (see section 3.3) and hence
inflation increases both of them over time. After a period of time in
which i is the effective rate of inflation, a and 0 will be increased to
a(1 + i) and ß(1 + i) respectively. Tables (3.11) to (3.17) show that
the mean and the standard deviation of the claim amounts have increased
from one period to the next.
3.9.2 The Prediction Technique
For our accidental damage claims, investigation shows that nearly
70% of the total cost of claims are reported and settled within the actual
period (quarter) of accident. Another 25% are settled by the end of the
second quarter. Therefore, it is reasonable to assume that all AD claims
are settled in the middle of the period of accident. Hence, if the
distribution of claim amounts during a period A, for instance, is 2
LN(x; u, a ), their distribution during a future period B will be
LN(x; p+o, a2) where it is assumed that the rate of inflation from the
middle of period A to the middle of period B is i and that d= log(1 + i).
If sizeable parts of the total cost of claims are settled over various
periods and, furthenrorc, if they take a] ong time after the end of the
period of accident to be se. ttied (e. g. for the third party liability
79
claims), then the rate of settlement irrst be taken into consideration.
A co, bined rate of inflation and settlement will be required for
prediction purposes.
Other factors such as seasonality may be affecting the nwrber and
amount of claims during different periods of the year. For instance, in
autumn and : winter, due to the bad weather conditions, it is likely to
have more accidents and more claims for larger amounts. The data of
tables (1.1) to (1.7) show that the number of claims for the 4th quarter
of the year is greater than for the other quarters. To nullify the
effects of such factors we suggest that each period of accident should
be used to predict the distribution of claim amounts during the same
period but in a future year(s). For instance, the model for the first
quarter of 1974 should be used to predict that for the first quarter of
1975 , etc.
We can use our available AD data to test this technique for three
different periods of accident, namely, we can use
73/4th quarter to predict 74/4th quarter,
74/1st quarter to predict 75/1st quarter,
and 74/2nd quarter to predict 75/2nd quarter.
The question remains as to what should be the appropriate rate of inflation.
It would be advantageous if we could show that changes in the parameters
of the model, over time, correspond to changes in the value of a particular
index of prices or wages. We shall consider three such indices
1- Me General Index of Retail Prices.
2- The Index of Average Earnings, Miscellaneous Services.
3- The Index of Motor Vehicles Repair Costs.
indices 1 and 2 are published in the "Monthly Digest of Statistics"
which is a publication of the Central Statistical. Office. Two of the
constituents of index 2 are the wages of rotor repairers and garages.
80
Index 3 is based on the experience of the Royal Insurance Company and is
published by the Economic Advisory Group which is part of the British
Insurance Association. The value of this index at the beginning of evti
quarter of the year is given. By taking the average of the index values
at the beginning and at the end of every quarter we can find the average
index for that particular quarter. For indices 1 and 2 we take the
average of the monthly values as the value of the index for each quarter.
The percentage change in the value of an index from-one quarter to a
future quarter indicates the rate of inflation to be used for prediction
purposes. After we have predicted the distribution of claim amounts for
one of the quarters mentioned above we can perform goodness-of-fit
tests between the predicted and actual distributions. A consistently.
good fit for different quarters will not only show the success of the
prediction technique but will also indicate the particular index which
should be used when predicting the distribution of accidental damage
claim anbunts. Naturally, we have to use a forecast value of that index
for periods in the future.
3.9.3 Prediction for the AD Data
To test the prediction technique, for the two-parameter lognormal
model, we wrote computer program P8. We considered it important to see,
first of all, what the predicted distribution would be like if inflation
was ignored. For the three quarters mentioned in the previous section,
the program was used with i=0. The results are presented in tables
(3.18) to (3.20). Let us look at table (3.18) for 74/4th quarter. The
program has produced a comprehensive table. The details of the predicted
quarter, the quarter used for predicticn and its parameters as well as
the rate of inflation and the index used for its calculation are provided.
81
The prediction parameters are computed and given as well as the actual
ones. The n,.; ans and the standard deviations of the predicted and
actual claim axrount distributions are also given. The output format for
the rest of the table is similar to tables (3.10) to (3.17) and the
column headings are self-explanatory. From tables (3.18) to (3.20) it
is clear that when inflation is ignored parameter p is underestimated
and the predicted mean and standard deviation of the claim amount are
much smaller than the actual ones. Therefore, we should expect a major
disagreement between the actual and the predicted distributions.
This is shown by the large values of the Chi-square and total expected
loss statistics. The large positive values of the total expected loss, T,
indicate that the distribution is heavily underpredicted when inflation
is ignored. Tables (3.21) to (3.23) show the results when inflation,
according to the Index of Motor Vehicles Repair Costs, is taken into
consideration. The results are now better, 'but still not quite satisfactory,
as the large values of the Chi-square and T statistics indicate. Here,
the negative values of T show that the distribution is over-predicted.
Next we used the Index of Average Earnings. The results are given in
tables (3.24) to (3.26). They show an improvement over the previous
index. The differences between the predicted and actual parameters are
smaller than before but the Chi-square statistics still show significant
differences between the predicted and actual distributions. The absolute
values of T are smaller but still show a large over-prediction. For
_the_GeneralIndex of Retail Prices, the results are presented in tables
(3.27) to (3.29). This index has produced the most satisfactory results
amongst the indices considered. The prediction and actual parameters
are close to each other and have produced the smallest values for the
Chi-square and T statistics. The Chi-square statistics stil]. indicate
significant differences between the predicted and actual distributions.
82
The total expected loss statistics, T, still indicate over-prediction.
If we calculate the values of the Kolmogorov-F. anirnov statistic, D,
from each of the tables (3.27) to (3.29) and compare them with the
critical values of D in table (2.1), we shall find that respectively
P 0.18 ,p0.15 and p 0.02 (see chapter 2, section 2.8).
Therefore, on the basis of this test, the differences between the
predicted and the actual distributions are not significant for the 74/4th
and 75/1st quarters but are significant for the 75/2nd quarter. The
large values of the Chi-square statistics are due to large contributions
from one or two lower tail intervals. If it had not been for these
intervals, the Chi-square values would have been satisfactory too.
This again indicates that we should consider a modification of the model.
The prediction technique performs satisfactorily, especially when used
in conjunction with the General Index of Retail Prices.
All the indices considered resulted in over-prediction. In predicting
the claims cost, and hence in reserving, it is better to over-predict and
be safe rather than under-predict and end up in a loss. However, there
must be a balance between over-prediction and tying up too much funds as
reserves. Therefore, we would prefer the General Index of Retail Prices
to the Index of Motor Vehicles Repair Costs. Besides, the former
results in closer agreement between the predicted and the actual
distributions. This indicates that changes, over time, in the amounts
of accidental damage claims correspond to charges in the General Index
of Retail Prices. The major cost of any repair, in particular for serious
accidents and hence for large claims, is for the replacement of the
damaged parts rather than for the labour charge. Therefore, it seems
reasonable that -accidental damage claim amounts should be subject to
inflation according to an index of prices rather than wages. The
inflation rates calculated prom the Index of Motor Vehicles Repair Costs
83
are too high. This index has been calculated from the experience of
just one company and we can see that it does not agree with the
experience of the company which has provided our data. One possible
reason could be that the forwwer company may be insuring more expensive
or luxurious cars. In the future stages of the present work we shall,
therefore, only use the General Index of Retail Prices in our predictions.
Before concluding this section it is not out of place to comment on
the importance of our proposed total expected loss statistic, T.
Table (3.27), for example, shows that we have predicted the total claims
cost with a difference (or an error) of £16,844 which is 3.16% of the
total actual cost. This is easily understood in a way that, say, the
Chi-square or the Ko1n gorov-S-nirnov statistics are not. It is thus that
we recommend the use of this statistic in actuarial work.
3.10 The 3-Parameter Lognormal model
our accidental damage data are from a portfolio of comprehensively
insured private motor cars. There is a voluntary excess, of amount c
(where c, 0), on these policies. For every claim of amount x+c, the
insured pays the amount c and the insurance company pays the amount X.
Therefore, it is reasonable to assume that the logarithm of the total
amount of claim, i. e. log(X + c), and not just log X, is distributed
normally.
In a portfolio, c nay have several different values. Our data
consists of payments by the insurance company only and the value of c
for every claim is not known. We assume that c is the same for all
policies and treat it as an unknown parameter. Hence, we are faced with
a 3-parameter lognormal distribution. This was defined in section 3.2
and some of its properties were given in section 3.4.
84
3.11 Test of Lo7ormality: - The 3-Parameter Case
3.11.1 The Graphical Test
It is useful to have a procedure for testing whether a given sample
of data is from a 3-parameter lognormal population. Aitchison and Brown
(1957) mention a graphical method of roughly estimating the unknown
location parmpeter c. We suggest using this method as a test of 3-
parameter legnormality.
From equation (3.4-6) we know that if the random variable X is distributed
2 as LN(x; -c , u, a ) then the locus of the points (log(x + c), z) is the
straight line
Zß log(x+ c) -0
Let us assume that we are given a sample of observations from a 3-
parameter log o=: al distribution. If the value of c was lcno wm, then the
set of points (log (x + c), z), calculated from the sample, should lie almost
on a straight, line (line 1 in figure (3.6)). If o is unknown, and we.
Z
Figure (3.6) - Straight line plot for a sample from the 3-parameter lognormal distribution
underestimate it as c' (i. e., use c' < c) then the points will show
a marked curvature in the lower tail values of x (curve 2 in figure
(3.6)). If we over-estimate c (i. e. use c' > c) then the curvature
would be in the opposite direction (curve 3, figure (3.6)).
Therefore, as a test of 3-parameter lognormality for a given sample of
data, we can initially plot the array of points (log x, z). If the sample
is from a 3-parameter lognormal distribution, its graph will resemble
curves 2 or 3 as above.. According to the shape of the graph we choose a
value of c in the required direction towards the straight line. We
plot the array of points (log(x + c), z) and, judging by the resulting
graph, accept c or modify it again in the required direction. If a
value c is found which makes the points (log(x + c), z) lie approximately
on a straight line, we will be satisfied that the sample is from a 3-
parameter lognormal distribution. This c can be considered as an estimate
of the unknown location parameter of the distribution. By using this
estimate we can then, as in the 2-parameter case, find estimates of 2
" and a.
3.11.2 Testing the AD Data for 3-Parameter Lognormality
Figures (3.1-a) and (3.1-b) showed that in the seven AD samples,
the points (log x, z) deviated markedly from the straight line for
the smaller values of x. We can now look at these plots and suggest
that the correct model is the 3-parameter lognormal. They all show' that
c=0 has, underestimated c and that some c>0 should be found to make
the points tend towards the straight line.
We wrote computer program P9 to plot the array of points (log(x + c), z)
for values of c=0,10,15,20,25. To find these points from the sarple
the procedure of section 3.6.1 and formula. (3.7-27) were used. The
plots for the seven samples of data are presented in figures (3.7-a)
86
"i, ire (3.7-a)
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we are satisfied that our samples are from 3-parameter lognormal distribu-
tions whose location parameters c have values between 10 to 20.
3.1.2 Estimation of the Parameters of the 3-Parameter Lognormal
Distribution
The existence of the unknown location parameter c, in addition to 2
and a, makes the estimation problem more difficult. Aitchison and
Brown (1957) do not deal with this problem for grouped data and only
examine the case when values of the individual observations in the
sample are given. They report methods of moments and quantiles where
three moments or quantiles, respectively, of the sample are put equal
to their corresponding population values. The method of maximum
likelihood is also considered which results in a non-linear equation in
c. They also report Kerrsley's (1952) method which consists of equating
the mean and two quantiles of the sample to their corresponding population
values. We can make assumptions, similar to the ones we made for the
two-parameter case, and modify the above methods for use with grouped
data (see section 3.7). However, as the analysis in the two-parameter
case showed, these methods would not provide very satisfactory results.
Their solutions may be of use, for example, in yielding initial values
for the parameters to be used in an iterative method of estimation.
Therefore, we will not consider these methods here.
Johnson and Kotz (1970) report the methods mentioned in Aitchison and
Brown (1957) and they do not consider estimation from grouped data. In
actuarial literature, Ferrara (1971) uses the method of quantiles to
87
bb,
determine c first and then finds u and c by least squares regression
(as in the 2-p ar=, eter case).
In this section, we will examine two methods which are directly
suitable for estimation from grouped data. The first one is the least
squares regression method. We will suggest a procedure for finding the
parameters which avoids solving non-linear equations. The second method
is the multinomial maximum likelihood which produced most satisfactory
results in the two-parameter case. This method will be modified and
used for the 3-parameter case. As-we mentioned in section 3.7.7, Tallis
and Young (1962) consider this method but they suggest equating the
partial derivatives of the loglikelihood function, with respect to the
parameters, to zero and solving iteratively the resulting set of non-
linear equations. This is more laborious than our proposed procedure.
3.12.1 The Method of Least Squares
Let us assume that a sample of grouped data as defined in section
3.7 is available. From section 3.4 we know that for the LN(x; - c, u, Q 2
)
distribution relationship (3.4-6), i. e.,
zq 1 log(xq + c) -Q
holds. If the random variable X (the claim amount) is assumed to be 2
distributed as LN(x; -c pa ), we can use the least squares regression
technique to find the estimates of the parameters. This consists of
minimizing SSD (the sum of the squares of deviations) simultaneously
with respect to c, u and a where, k-1
" SSD = [Zi -Q log(xi + c) + ý' l (3.12-1) i=1 c
One way to solve this problem would be to equate the partial derivatives
of SSD, with respect to c, p and a, to zero. Dut this will result in a
88
hlh. ý
system of non-linear equations in the parameters which has to be laboriously
solved by an iterative technique. We, however, suggest a different
method. Instead of searching for the set of parameters at which SSD
attains its minimum, we propose finding the minimum value of SSD and AAA
taking its corresponding estimates of the parameters (c , u, c ) as the
required solution. Let us explain this method in nore detail.
If our data is from a 3-parameter lognormal distribution, then
there exists ac at which SSD attains its minimum - see figure (*) below.
SSD
Figure (*) - The Graph of SSD with respect to c.
If we let c= co where co is some known value then the values of
log(xi'+ co), for all i, will be known and hence (3.12-1) becomes
similar to the SSD for the 2-parameter lognonnal distribution. Therefore,
the values of uo and vo corresponding to co can be found by the least
squares regression technique described in section 3.7.6. It will then
be possible to calculate the value of SSD, from (3.12-1), -corresponding
to (co' uo' ao). By choosing several different values for c and
calculating their corresponding SSDs we can find a range of values of c,
say (c1, c2), in which the SSD changes from a decreasing function to an
increasing function. The mininram of SSD lies in this range. By taking
89
C1 C C2 C
d, the required degree of accuracy for c, into consideration, we can
choose suitable values of c in (cl, c2) and hence reduce the range
gradually to (c ,c+ d) where SSD attains its minimum value at c 22
The set (c p, a) will be the required least squares estimates of
(c, ti, a) respectively. This technique can be programmed on the computer
and will then be much simpler and faster than solving a system of non-linear
equations.
Any prior knowledge about the maximum value of c should be utilized
in choosing the initial range of c: For example, in the case of our
accidental damage data we know that c, the voluntary excess, is at most
M. Therefore, to examine the behaviour of SSD, we should initially try
c=0,5,10,15,20,25 and 30.
3.12.2 The Multinomial Maximum Likelihood Method
This method was considered for the two-parameter lognormal
distribution in section 3.7.7. In this section we will present
the likelihood function for the 3-parameter case. Let us again assume
that a sanpie of grouped data as defined in section 3.7 is available. 2
If x is assumed to be distributed as LN(x; -cuo), by adopting the
notation of section 3.7.7,
22 pi = LN(xi; -c Ili Icy )- LN(xi-1; -c ,u ,Q)
log (x1 + c) -u log (xi-1 + c) -u pi 'N(a; 0,1) -N(Q1 001)
Therefore- the sample likelihood function will be proportional to L where,
L Pi i=1
and the loglikelihood function will be
90
k log L= nl log [ N(zi; 0,1) - N(zi-1,0,1) ) (3.12-2)
i=1
log(x1 + c) -u where zl =a
We will adopt the technique, described in section 3.7.7, of maximizing 2
log L with respect to (c, u, a ). To start the iteration process we can
use the least squares, or any other, estimates of the parameters. The
set of estimates (C, 11 $a ) which maximizes log L will be the required
multinomial maximum likelihood estimates.
3.13" Application of the 3-Parameter Lognormal Model to the AD Data
We wrote computer program P10 to use the least squares method to
estimate the parameters of a 3-parameter lognorml distribution
from a sample of grouped data. The program was run on an interactive
terminal which-made the task of estimation even simpler and faster. It
was used to estimate the parameters for our seven samples of accidental
damage data which were presented in tables (1.1) to (1.7). We explain
the procedure for a particular sample, say, 73/4th quarter. After the
sample data has been supplied to the program, it will require a value for
c along with an integer 'IPRINT' which is either 0 or 1. IPRINT =0
indicates that only the value of SSD is required. IPRINT =1 indicates
that a comprehensive table of results is also required. The results of
the computer rum to find the estimate of c for 73/4th quarter data are
presented in table (3.30). We started by c=0 and increased c in steps
of S. For each c its corresponding SSD was printed. We noticed that in
the range (c = 15, c= 25) SSD changed from a decreasing function to an
increasing one. Therefore, as a next step we tried c= 18 whose SSD
shoved that the optimum c is in the range (18,25). Because an
91
accuracy of 1 is acceptable in the estimate of c, we then tried c= 19
whose SSD indicated that the optimum c should be in the interval (20,25).
We then tried c= 21 whose SSD was less than that of c= 20. Therefore,
the optimum range was reduced to (21,25). As the next step we tried c= 22
whose SSD was greater than that of c= 21. Therefore, with our required
accuracy, c= 21 is the least squares estimate of c. This value was then
-supplied to the program, with IPRINT = 1, and the-estimates of u and a2
along with an extensive table of results were produced (see table (3.31)).
Similar tables were obtained for other samples and the results are
presented in tables (3.32) to (3.37). We can see that the value of c
for our data ranges from 13 to 25 which is reasonable. has generally
increased over time from 4.78 to 4.98 while v2 ranges from 0.67 to. 0.77.
The means and standard deviations of the fitted rndels are very close to
the means and standard deviations of the samples. The Chi-square statistics
are smaller than in the 2-parameter case but still indicate significant
differences between the fitted distributions and the actual sample values,
except in the case of 74/4th, 75/ist and 75/2nd quarters where the X2
values are not significant. The total expected loss statistics are very
small and are at most 2% of the total actual cost. This indicates an
overall agreement between the fitted models and the samples. It was
important to see how closely the actual sample points lie on the least
squares straight line. Therefore, we wrote computer program P11 to plot
the sample points (log(xi + c) , zi) and the least squares line
Z. log(x + c)
For the seven AD sau les the graphs are presente d in figures (3.8-a)
and (3.8-b). The points, for each sample, generally lie closely on the line.
92
In the lower tails the fits are very good, unlike the large deviations
which were observed in the 2-parameter case. It is encouraging to see
that the introduction of parameter c into the model has produced more
satisfactory results.
-Computer program P12 was written to estimate the parameters by the
nethod of mul tinomrdal maximum likelihood. This program was used with the
samples of accidental damage data. z
estimates of (c, p, Q ) were supplied,
For each sample, the least squares
to the program, as the starting
values for the iteration process. The results for the seven samples are
presented in tables (3.38) to (3.44). In each case the estimates of the
parameters, the mean and the standard deviation of the fitted distribution
and an extensive table have been produced. The estimates of c range from A 10 to 21 with an average of about 15 which is reasonable. The values of u
generally show an increasing pattern and are in the 4.65 to 4.91 range.
a2 ranges from 0.71 to 0.86. We can observe that the Chi-square values
are smaller than their corresponding values for the least squares method.
They do not indicate significant differences, except for 74/1st and 74/3rd
quarters, between the fitted distributions and the sample values. For
74/1st and 74/3rd quarters the differences are, respectively, highly
significant and significant, but we should note that the major contribution
to the X2 values comes from just one cell in each case. The total
expected loss statistics are small and are at most 1.2% of the total actual
cost of claims. This indicates a very satisfactory general agreement
between the theoretical models and the samples values. Values of the
Kolmogorov-Srurnov statistics for all the samples were calculated, by
using the. (Actual-Expected) column in each table, and were compared with
the critical values of this statistic as given in table (2.2). Only for
74/1st quarter this statistic indicated almst significant differences
between the fitted distribution and the sample values. In all other cases
93
the differences were satisfactorily not significant.
The least squares method is very easy to use but, as in the case of
the 2-parameter lognormal, the multinomial maximum likelihood method has
produced better results in terms of the goodness-of-fit test statistics.
We, therefore, again recommend the use of this method in estimating the
parameters of the distribution from grouped data.
In order to see the agreement between the frequency curve of the
fitted distribution and the histogram of the sample values we plotted then,
for each of our seven AD samples, by using a modified version of program
P7 to suit the 3-parameter lognormal model. The resulting graphs are
presented in figures (3.9.1) to (3.9.7). For each sample, the rrultinomial
maximum likelihood parameters have been used to plot the frequency curve.
Each figure represents two graphs. One for claim amounts < £600 and the
other for claim amounts > £600. The agreement between the frequency curve
and the histogram is very satisfactory for all samples. The curve
portrays the mode of the histogram. The large deviations in the lower
tail between the curve and the histogram which were observed in the 2-
parameter case (see figures (3.5.1) to (3.5.7)) have now disappeared.
The long tail of the histogram has also been modelled very well by the
distribution. We are, therefore, satisfied that the 3-parameter
lognormal distribution offers a correct model for our accidental damage
claim arrounts. Hence, we shall next consider the problem of predicting
the distribution of claim amounts during a future period.
94
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2 '-' 0 141.1. ) 1 W. U I'liv,
xie-t S. el
4.51
3.5c
3.
2. '
2.1
1 .:
t..
Fiuio (3.9.6)
ISTtGRA1 _CF 75/1ST QUART[: R DATA AND 3-PAR.
DCNCRNAL P. C. F. WITH MULTN. MAXL. PARAMETERS
-21.0 MEN'-1.915 SIGMA S0. =0.712
27(", r) : iýC f.! 1 Dili ý~ý `C'i4
C! V "Cid
cjcir IWO
I 2l f1
T.
7
i
i
2
1
1
1
1
1
1
1
Figure (3.9.7)
f"f
X1o-i 5. C
9 . 51
i. el
2.51
2.
1.
1.
CORAM OF 7'-'2ND CUAPTER DATA AND 3-PAR.
IORflAL P. D. F. WITH IULTN. MARL. PARAMETERS
;. e t1 W-4.072 SIGMA S0. =7. C7
ý'ýýýýi 1ýrrr ýrýr"Yr ý: i:
ý; {i : li'1, ; "; c; s., f; C; s"s+c1 . ; 'C'sý . ;: ßs1
3.14 Prediction of the Claim Amount Distribution :- The 3-Parameter
Lognormal Model
The importance of predicting the future cost of claims was mentioned
in 3.9 and here we will adopt techniques similar to those of that section
for predicting the future distribution of claim amounts. To save us
repeating some of the arguments of that section we suggest referring back
to it at this stage.
3.14.1 The Effects of Inflation on the Parameters of the Model
Let us again assume that the effect of inflation on a claim of amunt
X, during a period of time, is to increase it to U= X(1 + i) where i
is -the rate of inflation, according to some appropriate index, for that 2
period. If X is assumed to be distributed as LN(x; c, u, a ), then by a
transformation of variables we can show that U= X(1 + 1) will be
distributed as : 2
IN(u; rc, u+d, a) where r=1+i and d= log(1 + i)
Therefore, when the distribution of X is known, we can modify its parameters
to obtain the distribution of claim amounts in a future period. The
foregoing shows that inflation increases both c and I. in time but does not 2
affect a. This supports our intuition that the amount of voluntary
excess, c, should go up in time due to the inflation. For our accidental
damage data, the increase, over time, in the values of u is obvious from
tables (3.38) to (3.44). The values of c for the later samples are also
larger than for the earlier ones.
Based on the results for the two-parameter case, we suggest that the
technique of section 3.9.2 be used for predicting the distribution of
accidental damage claim anounts during a future period. The appropriate
index proved to be the General Index of Retail Prices and wo will use it
here again.
95
3.14.2 Prediction for the AD Data
To test the prediction technique on the accidental damage data,
we modified computer program P8 for use with the 3-parameter model. For tl
reasons presented in section 3.9.2, we used the parameters of the distri-
butions for 73/4th, 74/ist and 74/2nd quarters to predict the
distributions of claim amounts in 74/4th, 75/1st and 75/2nd quarters
respectively. The appropriate rate of inflation in each case was
calculated from the General Index of Retail Prices, as described in
section 3.9.2. The results, in the form of extensive tables which allow
comparisons between the actual and expected (predicted) distributions, are
presented in tables (3.45) to (3.47). In all three cases the Chi-square
statistics are relatively small, considering the amber of degrees of.
freedom, and do not indicate any significant differences between the
predicted distributions and the actual sample values. The total
expected loss statistics are small and show an over-prediction of the
total cost which is more satisfactory than an under-prediction. They
are a small proportion of the total actual cost in all cases. The
number of changes and non-changes of signs in the 'A - E' (i. e. , the
actual minus expected frequencies) column are roughly equal. The
Kolmogorov-Smirnov statistics were calculated from the results tables and
in each case were compared with the critical values given in table (2.1).
They showed that the differences between the actual and predicted
distributions were not significant for 74/4th and 75/1st quarters but
were almost significant for 75/2nd quarter.
These results show a great improvement over those given in tables
(3.27) to (3.29) for the 2-parameter model. The consistent satisfactory
outcomes of the goodness-of-fit tests indicate the success of the
prediction technique and the appropriateness of the General Index of Retail
96
prices for inflating accideiLal damage claim amounts. Various assumptions
about the future rate of inflation or values of the parameters can be
tested by this technique and the computer program associated with it. A
value for the future rate of inflation and a set of parameters can be
adopted as standard values and other rates and sets of parameters can then
be tested against them. The total expected loss statistic, T, in
each case will indicate, in terms of the easily understood amount of
coney, by how much the assumptions differ frone each other.
It may appear simpler just to increase the actual mean claim amount
in a particular period by the effective rate of inflation between that
period and a future one and to take the result as the mean claim ammunt
for the latter period. Using the total number of claims we can than
calculate the total expected cost of claims for the future period. But,
it can be shown that this method will result in a much larger over-
prediction of the total claims cost than by our distribution theory
approach. For example, the sample mean claim amount for the 73/4th
Quarter is £150.36 and the effective rate of inflation from this period
to 74/4th quarter is 18.2% which yields a mean claim amount of £177.7 for
the latter quarter. The number of claims for that quarter is 3064
and hence the total expected cost would be £544,551 which shows-an over-
prediction of £10,884, as compared with £4,093-by our approach. The
former discrepancy is 2.03% of the total actual cost while the distribution
theory resulted in a ratio of 0.77% only. Similar results can be observed
for the other two quarters we have considered. Therefore, one of the
merits of our distribution theory approach is that it results in accurate
but smaller reserves. This means that the insurance company will not
need to tie up funds in unnecessary reserves.
97
3.15 Conclusions
In this chapter the two and three parameter lognormal distributions
were studied as models for the distribution of claim amounts, in general
insurance and in particular for motor insurance accidental damage claims.
Simulation exercises showed that the graphical method is a good way of
testing for lognormality. In the 3-parameter case, this test also
provided a rough estimate of the location parameter c. It was observed
that, even in large sanpples of (size 2500) observations from an actual
lognormal population, the points (log x, z) deviate markedly from the
straight line for values of x in the upper tail of the distribution.
Therefore, in testing exercises, we should expect to find that the points
in-the upper tail deviate from the straight line which may fit the
rest of the distribution.
Several methods of estimation for the 2-parameter case were considered
and their efficiencies were calculated. It was observed that the
approximate maximum likelihood method was the most efficient amongst them.
It was mentioned that the most efficient mathod for-estimation from
grouped data is the multinomial maximum likelihood method. A teclmique
for estimating the parameters by this method was proposed and shown to
provide better estimates (in terms of a closer agreement between the
model and the actual data). For the 2-parameter case, the approximate
maximum likelihood and the multinormal maximum likelihood methods gave
approximately the same results and hence the use of the former method in
this case should be recormiended. In the 3-parameter case our proposed
technique for estimating the parameters by the least squares method was
shown to be very sinVle when programed on the computer. Ho4; ever, the
results produced by this method were not as satisfactory as those of the
multinormal maximum lir. eli. hcod method.
For the accidental damage data, it was found that the 3-parareter
J8
form of the distribution is the appropriate model. The inflation was
shown to be the main cause of the increase of the claim an, t over tim:..
We proposed a technique for predicting the distribution of the claim
amounts in a future period. This involved updating the parameters c and
j; with respect to an appropriate rate of inflation. The General Index
of Retail Prices was shottin to provide the appropriate rate of inflation
for accidental damage claims. We tested the prediction technique on
data from the past and it proved to be very successful.
The total expected loss statistic, T, was suggested in chapter 2 as. a
goodness-of-fit measure. In the present chapter it was put to use and
shown to-provide an easily understood measure (in monetary terns) of
overgraduation or undergraduation. Its use is particularly recommended
in prediction exercises where it indicates the extent to which a model
is overpredicting or underpredicting the actual total cost of the claims.
3.16 Tables
99
Cocfficicnts of skewness (gl) and excess kurtosis (g2)
and their standard errors for the distribution of log X, where X= claim amount
Table (3.1)
Accidental Damage Samples
Period of accident
gl Se(g1 ) g2 Se(g2)
73/4th quarter -0.107 0.044 -0.506 0.089
74/1st " -0.076 0.049 -0.503 0.099
74/2nd' -0.144 0.050 -0.464 0.100
74/3rd -0.179 0.046 -0.352 0.092
74/4th -0.199 0.044 -0.428 0.088
75/1st " -0.237 0.048 -0.394 0.096
75/2nd -0.184 0.049 -0.376 0.098
Table (3.2)
Simulated Samples
Sample g1 g2
1 -0.049 -0.328 2 0.105 -0.143 3 0.065 -0.167 4 0.122 -0.122 5 -0.007 -0.170 6 -0.013 -0.209 7 -0.037 -0.412 8 -0.039 -0.369 9 0.030 -0.223
10 0.067 -0.272 For all samples: Se(gl) = 0.049
Se(g2) = 0.098
100
Table (3.3)
Efficiencies of various methods of estimation for the
Two-Parameter Lognormal Distribution
Method of estimation Eff (ý) Eff (v2) Eff (a) Eff (ß)
Moments 0.044 0.095 4.145 17.710
Approx. Max. Likelihood 0.021 0.036 3.944 9.768
7& 93% quantiles 0.131 0.188 7.267 24.493
27 & 73% quartiles 0.026 0.050 4.634 12.186
Median & Coeff. of variation
0.026 0.095 7.525 24.561
Least squares regression
0.051. 0.104 4.363 18.999 1
101
Table (3.4-a)
73 /4TH QUARTER ***#*
FITTING THE TWO PARAMET ER LOGNORMAL DIS. BY DIFF. METHODS.
TABLE OF ESTIMATED PARAMETERS AND STAT ISTICS
METHOD--> MAXIMUM 7&93 ý% 276,73 % MOMNTS LIKELIHOOD QUANTILES QUANTILES GRAPHICAL RECRESSION
STATISTICS
MEWHAT 4 . 583 4.503 4 "325 4 "5 10 4.554 4.568
SIGMA2H'AT 0.860 1.089 1.321 1.018 0.893 0 F176
ALPHA=MEAN 150.359 , 155.669 146.318 155.935 "148.495 149.333
BETA=S. D. 175.482 218.530 242.476 207.387 178.399 176,791
COF OF VAR 1.167 1.404 1.657 1.330 "1.201 1.184
MEDIAN 97.832 90.315 75.596 93.713 95.000 96.363
MODE 41.417 30.400 20.179 33.846 38.882 40.125
CKE -INESS 5.091 6.978 9.523 6.342 5.338 5 . 21 1
" KURTOSIS 71.228 153.819 341.219 121.219 79.719 75.2613
102
Table (3.4-b)
## 73/4TH QUARTER **##*
EXPECTED NO. OF CLAIMS BY DIFF. MET HODS OF ESTIMATI ON
T WO PARAM ETER LOGNORM AL DIS.
AMOUNT f ACT. NO. MOMNTS MAX. LIK. 7(93°JQS 27f*73%Q S GRAPH. REG. 1- 30 478. 318. 454. 654. 405. 349. 334.
31- 60 518. 602. 613. 634. 607. 615. 609. 61- 90 461. 501. - 458. 423. 469. 496. 499. 91- 120 .
359. 373. 329. 290. ' 342. 365. 369. 12 1- 150 239. 274. 240. 207. 250. 266. 270. 151- 180 213. 203. 179'. ' 153. 187. 197. 200.
" 151- 210 148. 153, 136. 116. 142. 148. 150. 211- 240 102.
- 117. ' 106. 90.. 110. " 113. 115.
241- 270 Al. 90. 83. 71. 86. ß?. . 89. 271-. 300 58. 71. 67. 57. 69. 69. 70. 301- 330 66. 56. 54. 47. 56. 55., - 55. 331- 360 45. 45. 44. " 35. a5. 44. 4t. 361- 390 39. 37. 37. 32. " 37. 36. 36. 391- 420 35. 30. 31. 27. 31. 29. 30. 421-"450 34. ' 25. 26. 23. 26. 24, 24. 451- 480 20. 21. 22. 20. 22. 20. 20, 451- 510 '29. 17. 19. 17. '19. 17, 17. 511- 540 1Q.. 14. "16. 15. 16. 14.. 14. 541- 570 8.. 12. 14.
_13.. 14. 12. 12.
571- 600 9. 10. 12. 11.. 12. 10. 10. 601- 700 29. 25.
. 30. 28. 30. " 25. 25.
701- 800 18. 16. 20. 19. 19. 16. 16. 801- 900 20. 10. 14. 14. 13. 10. 10, 901-1000- 6.. 7. 10. 10. 9. 7. 7.
1001-1100 4. 5. . 7: 7. 7. 5. 5. 1101-1200 4. 3. 5. 6. 5. 3. 3. 1201-1300 1. ?. 4. 4. 4. 3. . 2. 1301-1400 3. 2. 3. 3. 3. 2. 2. 1401-1500 1. 1. 2. 3. 2. 1. 1. 1501-1600 0. 1. 2. 2. 2. 1. 1. 1601-1700 1. 1. 1. 2. 1. 1. 1. 1701-1800 (1. T. 1. 1. 1. 1. 1. 1801-1900 0. 0. 1. 1. 1. 1. 0. 1901-2000 1. 0. 1. 1. 1. 0. " 0. 2001-2100 0. 0. 1. 1. 1. 0. 0. 2101-2200 0. 0. 1. 1. 0. 0. ' 0. 2201-2300 . 0. 0. "
0. 1. 0. 0. 0. 2301-2400 1.. Q. 0. 1. 0. 0. U.
TOTAL 3045. 3043. 3043. 3043. 3044. 3013. 3042.
103
'. Cable (3-4-0
##### 73/4TH QUARTER ## #,
(ACTUAL - EXPECTED) CLAIM NO. S BY. DIFF. METHODS
TWO-PARAMETER LOGNOR! 'AAL bIS.
AMOUNT £ MOMENTS MAX. LIK. 7F, 93%QS 27G73%QS GRAPHCL AEG 1- 30 160. '
24. -176. 73. r 1 . 144 31- 60 -ý 4. -95. -116. -119. 97, -97.
. -91 61- '90 -40: 3. 38. -El. . -35.
. -3l. 91- 120 -14. 30. 69. 17. -6. -10. 121- 150 -35. -1,, 32. -1 1. -27. -31 151- 181) 10. 34. 60; 20. 16, ,
13. 181- 210 -5. ' 12. 32. 6. 0. -2. 211- 240 -15. -4. 12. -8, -11. -13. 241- 270 -9. -2. 10, -5. -7ý -8, 271- 300 -13. -9. 1. -11. -11. -12. 301- 330 10. 12. " 19. 10. 11. 11. 331- 360 0. 1. 7. 0. 3. 361- 390
' 2. 2. 7. ' 2. 3. 3. 3. 20 391= 4 5'. 4. B.
421- 450 9. H. 11. 8. 10. 'ý0 . -ý
451- 480 -1. -2. , 0. -2. 0. ' 0
\ \ 481- 510 12. 10. 12. 10. 12. . 12. 511- 540 .. 0.. -2. -1. -2. 0. 0. 541- 570 -4. -6, -5" -t, " -4. - 4.
... 571- 6001 -1. -3. -2. -3. -1, . -1 601- 700 4. -1. 1. -1. 4. . 4 701- 800 2. -2. -10 -10 2. . 2 801- 90-0 10. 6. 6. 7. 10. .
10. 901-1000 -1. -4. -4. -3. -1. -1 1001-1100 -1, -3. ' -3. -3.
. 1 101-1200 1, -1 . -2. --1 " 1, 1, 1201-1300 -1, -3. -3. -3. -2. -1, 1301-1400 1. 0, 0.. n. i. 1'
' 1401-1500 0. , -1 , -2. �1. 0. . 0
1501-1600 -1. -2. -2. -2. -1, .
. -1 16111-1700 0. 0, -1, 0. 0 0 1701-1800 -1. -1ý
' 1801-1900 0; -10 0 1901-2000 1. 0. 0 '
1 2001-2100 0. -1. -1, -1, 0, . 0 2101-2200 0. -1. -1. 0. 0 , C 2201-2300 0, 0. -1. 0. . 0. . 0 2301-2400 1. 1. 0, 1" 1. . 1.
10^
14
a
a
a
Table (3.4-d)
73/4TH QÜARTE'R ##x#*
" CHI SQ. GO ODNESS OF FI T- TEST FOR DIFF. ME THODS :.
T WO-PARAMETER LOGNORMAL DIS.
AMOUNT $ MOMNTS MAX. LIK. (A-
7ET93%QS E)**2/E
27&73%QS GRAPHCL REG.
1- ...
30 ....... 80.503
........ 1.2 9
.. 47.364
......... 13.158
........ 47.682
......... 62.014
31- 60 11.721 14.723 21.224 13.049 15.299 13.598 61- '90 3.1,94 0.020 3.414 0,136 2'. 470 2.894 91- 120 0.525 2.736 16.417 0.845 0.099 0,271
121- 150 4.471 0.004 4.947 0.484 2.741 3.559 151- 180 0.493 6.4513 23.529 3.615 1.299 0.845 181- 210 0.163 1.059 8.828 0,254 0.000 0.027 211- 240 1.923 0.151 1.600 0.582 1.071 1.470
241- 270 . 0.900 0.048 1.408 0.291 0.557 0.719
271- 300 2.380 1.209 0.018 1.754 1.754 2.057
301- 330 1.786 2.667 7.681 1.786 2.200 2.200
331- 360 0.000 0.023 1.. 289 0.000 0.023 0.000 361- 390 0.108 0.108 1.531 0.106 0.250 0.250 391- 420 0.833. 0.516 2.370 0.516 1.241 0.833 421- 450 .
3.240 2.462 5.261 2.462 4.167 4.167
451- 480 0. Cet A 0.1132 0.000 0.182 0.000 0.000
4131- 510. 8.4-/1 5.263 8.471 5.263 8.471 8". 4 71
511- 540 0.000 0.250 0.067 0.250 0.000 0.000
541- 570 1.333 2.571 1.923 2.571 1.333 1.333 571- 600 0.10r) 0.750 0.364 0.750 0.100 0.100 1 601- 700 0.640 0.033 0.036 0.033 0.640 0.6,10
701- 800 0.250 0.200 0.053 0.053 0.250 0.250
Sol- quo 10.000 2.571 2.571 3.769 10.000 10.000
901-1000' 0.143 1.600 1.600 1.000 0.143 0.143
DOTAL cxzsQ, 133.8 54.7 174.3 59.9 102.3 116.5
D. F. 24 26 26 25 24 24 P< 0.001 0.001 0.001 0.001
, 0.001 0.001
105
Table'(3.4-ej /
***-X* 73It1 TH QUA TER ## **
ACTUAL AND EXPECTED CLAIMS COST BY DIFF. METHODS
TWO PARAMETER LOGNORMAL DIS.
AMOUNT £ ACTUAL A40MNTS MAXLIK . 7&93%OS 27&730 QS GRAPHCL REG .' 1- 30 7409, 4.929. 7037. 1013?. 6277: 5409. 5177. 31- 60 23569. 27391. 27891. 28847. 27618. 27983. 27710. 61- 90- 3481)5. 37825'. 34579. 31936. 35410. 37448. 37674, 91- 120 37874. 39351. 34709. 30595. 36081. 38507, 38929,
121- 150 32384. 37127. 32520. 28040. ,
33875. 36043. 36585. 151- 180 35251. 33596. 29624. 25321. 30948, 32603. 33100, 181- 210 28934. 299.11. 2E5ß8. 22676. 27'761. ' 28934. 29325. 211- 240 23001. 26383. 23903. 20295. 24805. 25481, 25932. 241- 270 20695. 22995. 21206. 18140. 21973. 22464. 22740, 271- 300 16559. 20270. 19128. 16273. 19699. 19699. 19985, 301- 330 20823. 17668. 17037. 14829. 17668. 17352. 17352. 331- 360 15547. 15547. 15202. 13129. 1,5547, 15202. 15547. 361-'390 14644. 13893.
. 13893. 12016. 13893. 13518. 13518.
391- 420 14193. 12165. 12570. " 10948. 12570. 11759. 12165,. 421- 450 14807. 10887. -11323, 1.0016, 11323. 10452. 10452. 451- '180 93111.. 9775. 10241. 9310, 10241. 9310. 9310, 481- 511) 14369. 8423. 9414. 11423. 9414. 8423. 8423. 511- 540 7357. 7357. E408, 7882. 8408. 7357. 735 7. 541- 570 4444. 6666. 7777. 7222. 7777-. 6666. 6666, 57 1- 600 5269. . 5855.. 7026. 6440. 7026,
_ 5855. 5855.
601- 700 18864, 16262. 19515. 18214. 19515. 16262. 16262. 701- Bon 13509. 12008. 15010, 14259. 14259. 12008. 12009, 801--900 17010. 8505, 11907. 11907. 11056, 8505. 8505. 901-1000 5703. 6653. 9505. 9505. 6554. 6653. 6653,
1001-1100 4202. 5252. 7353. 7353. 7353. 5252: 5252. 1101-1200 4602. 3451. 5752. 6903. 5752. 3451. 3451. 1201-1300 1258. 2501. 5002. 5002. 5002. 3751. 2501 1301-1400 4051. 2701. 4051. 4051. 4051. 2701. .
" 270 1, 1401-1500 1450. 1450. 2901. 4351. 2901. 1450. 1450, 1501-1600 0. 1550, 3101.
. 3101, 3101. 1550. 1550, 1601-1700 1650. 1650. 1650. 3301. 165Q. 1650. 1650, 1701-1800 0, 1751. 1751. 1751. 1751. 1751, 1751, 1801-1900 0. 0. 1850. 1850. 1850. 1850, 0.
% 1901-2000 1950. 0, 1950. 1950. 1950. 0, 0. 2001-2100 0. 0. 2050. 2050. 2050. 0. 0. 2101-2200 0. 0. 2150. " 2150. 0. 0 0. 2201-2300 0. 0. '0. 22.50. 01 o, 0. 2301-2400 2350. 0. - 0. 2350. 0.
, o. 0.
....
TOTAL CL CAST ........
457843. ........
451757. ........
'165582. ........
43479;, .........
469117. ........
447327.
. ... ....
4.475'i 1, N
N
J
105
Table (3.4-i)
##* 73/4TH QUARTER ***+ý
EXPECTED LOSS BY DIFF . METHODS: (A-E)-(CLAIM AMOUNT)
' TWO-PARAMETER LOGNORMAL DIS.
AMOUNT £ MOMNTS MAX. LIK. 7&9301, -QS 27Ci73%QS - GRAPHCL REG,
1-- 30 240. 372. -2728. 1131. 1999. 2232. 31-" 60 -3822. -4322. -5278. -4049, -4413. . -4140. 61- 90 --3020. 227, 2869. -604. -2642. -2869 91- 120 -1477. 3165. 7279. 1793. -633-.
. -1055. 121- 150 -4742. -135. 4336. -1490, -3658, -4200. 151- 180 1655. 5627. 9930. 4303. 2648. 2151,
181- 210 -978. 2346. 6256. 1173. 0. -391. 211- 240 -3382. -902. 2706. . -1804. -2480. -2931, 241- 279, -2299. -511. 2555. -1277. -1788. -2044. 271- 300` -3711. -2569. 286. -3140. -3140. -3426. 301- 330 3155. 3786. 5994. 3155. " 3470. 3-1,70,
331- 360 0. 346. 2418. 0. 346, 0. 361- 390 751. 751. 2628. 751. 1126. - 1126. 391- 420 2027. 1622. 3244. 1622. 2433. 2027. 421- 450 3919-. 3484. 4790. 34134. 4355. 4355. 451- 480 -466. -931. . 0. -931. " 0. 0 481- 510 5946. - 4955. 5946. 4955. 5946, , 5946. 511- 540 %
0, -1051, -525. -1051, 0, 2 . 511- 570 -2222. -3333. -2777.. -3333. -2222. -2222. 571- 600 -585. -1756. -1171. . -1756, -585. -585, 601- 700 2602. -651.. 651.
. -651. 2602. 2602. " 701- 800 1501. -1501. -751. -751. 1501. ' 1501.
801- 900 8505. 5103. 5103. 5953. 8505. 8505, 901-1000 -951. -3802. -3802, -2851. -951. -951. 1001-1100 -1050. -3151. -3151. -3151, - -1050, -1050. 1101-1200 1150. -1150, -2301. -1150, 1150, 1150,
1201-1300 -1250. -3751. -3751. -3751. ' -2501. -1250 1301-1400 . 1350. . 0. 0. 0. 1350. ' . 1350.
1401-1500 0. -1450. "
-2901, -1450. 0. 0, 1501--16011 -1550. -3101. -3101. -3101. -1550.
. -1550. ` 1601-1700 0. 0, -1650. .
01 0. 0 1701-1800 -1751. -1751. ,. -1751. -1751, -1751. -1751. 1801-1900 0, . -1850. -1850,
-1850, -1850,. 0.
1901-2000 1950. 0. 0. 0. 1950. 1950 2001-2100- 0. -2050. -2050. -2050. 0, .
0 2101-2200 0. '-2150. -2150, 0. 0. 0. 2201-2300 0,, 0, -2250. 0. 01 p.
,. 2301-2400 2350. 2350. 0, 2350. 2350. 2350. .............
TOTAL .....
6(0,86, ..........
77739.
..................
2305.1. -11274. ........
10516, .......
10301. 1OTAL EXP. LOSS
--------------= 1.33% -"1.69'/1 5.03% -2.4f`ß, 1 2.30° 2.. 25^. 1'OTAL ACT. COST
I
107
Table (3.5)
ýti Kx 74 / 1: i T NtJAn'TER ** x**
FITTING 1 -HE TWO PARKMET ER L()GNE; HMAL 0: 16. BY DIFF. METHODS
"AHLE: [if: E5i :i MA TED PMHAIAE+ T EH S /AND SiAT : (ST': CS
METHOD--> FlAXINUM3 7i; c3 /1 27(, 73 ^l, MUPANIS l:: IKEI_: (H(l[l[) [-itlllý TILES ialýKýJ'i : it. Es GRAPHICAL. Fl(: GRESSICht
......... ........ ........ .... .......... .......... .......... ..
Sih TY. i. i?: CS
M"E1144 AT 4.5A9. 4.497 4.3112 4.51? 4.511 "4.581 "1
SX(zMA2HAT (1 . 942 1 . [IAA) 1.. 33A 0 97 3 0,975 (i . A53
ALPHA= FAN 149-F45-3 1 Eiu 599 15(I . (17f, 1110.995 140.125 1414.534
BETA=U. 1). 172 .l ()ä 216.901 251.666 191.132 1911-411 173.4 05
., - (: (JF OF VAR . 1.1a9 1.004 1.677 1.203 1.205 1.160
MEDIAN 98.371 09.707 76, A66' 91.603 91 . 1)00 97.620
MOUE 42.3Q 1 30 . 200 20.164 34 . 625 34 . 322 41.615
SKEWNESS , ý1f+5 6976 9.746 5.951. 5.977 5.042
KURTOSIS 67.1(15 153. x2 362.56() 104. ()21 104.779 64.626
7'C:, ".:. CH: SQ. 139.3 56.4 142.2 60.9 67o Q 128.2
D. ý'" 23 24- 24 23 23 23
p< 0.001 0.001 0.001 0.001 0.001 o. od1 T 3876 1197 20559 10161 11812 5391 T
," 1
0
108
Table (3,6)
7 4/2ND QUARTER . ##" x*
FITTING'THE TWO PARAME TER LOGNORMAL DIS. BY DIFF. ME THO'DS
TABLE OF ESTIMATE D'PARAMETERS AND STAT ISTICS
METHOD--> MAXIMUM 7', 93 °', 27F, 73 °% MOMNTS LIKELIHOOD QUANTILES QUANTILES GRAPHICAL REGRESSION
. """"""""""""""""" "a""""""""" """"""""""""""""r"""r """"""""" """""""""""
STATISTICS
1.1EWHA T 4.620 4.531 4.377 4.576 4.564 4.609
SIGMA2HAT" 0.801 1.057 1.306 0.9311 0.913 0.818
ALPHA=MEAN 151.574 --157.531 152.969 155.186" 151.505 151.201
BETA-S"'D. 167.980 215.867 250.932 193.447 1114.977 170.169
COF OF VAR 1.108 -1.370 1.640 1.247 1.221 1.125
MEDIAN 101.543 92.862 79.622 97.107 96.000` 100.430
MODE 45.572 32.269 21.572 38.023 38.544 44.308
SKEWNESS 4.686 6.684 9.335 5.677 5.48.3 4.802.
KURTOSIS 58.669 138.092 324.022 92.424 84.993 62.099
TOTAL (: N! SQ. 136.3 52.2 142"5 64.3 . 87.1 ' 119.3
-n" F. 25 24 25, 24 23 - 24
F< 0.001 0.001 0.001 0.001 0.001 0.001
T 3786 -4373 16604 -1293 4025 5616 T
Lý , ýC. '" CCST 1
'109
Table (3.7)
xk* 74/3P[) UUAP E! 1 *** .
"F7 º_NG YHE TWO OA' A ME TEH LOGNUHj! %[. [)): S. HY 01 FF . "A -f HUU
'IA(ILE OF ES"iXMAiE- [) PPRA, '4'r. 'iEH S H'f) 1; iAI 1h IC`i
}1E i H7i)__> ýMX: +: M1J 4 WI C19 7, 21 ; "):; *ýý Mtl'1N i.; LIKE: t. 1H1)(10 0U/tN ILE; ) GHAHH12AL r1E: G(? c.:;; 1!., '; '
ýfE:: HÄT a. 6U 4,622 llfiy 4 . 67/1
ii7; zMi 2HP T (I "R2 1.1162 1 , 253 0.020 fl, 71 t), ý; E, L+
1l6.1: aý1 6h 1f' 4,21?. 1E, A, (1: 44 1. '7fI. 6, i" 1'"5,, '1ý, n
i; E ýH=ci, iº. 9. f1.2 ? 3: '. )i, 11 ?q ; f03C)2 21,74 1U3.4 '7
(; [IF oof- VF. ý' 1.1 iý, 1, : i')5 1,5A 1 1 . 22'' I, 2P 1 1,1'72
! ";, )1F. CJ äi. 23'' 1(14. ` 1? 1(15. IV,?. 1K,
c' [! 7J ; i'+ ý? 2 4.22 n1 . 7ý, a r . 741; X15,133
týf. E ", i: E.:;;; tL"c'') h, 'ý? ý, ý. fýcý'I 5.1,40 P 5"12Fi
(4.2-,. i 14(1.2't 2h0"t, ý1 F'7,14: i 1cI: i. 51° 72.401
` TOTAL Cz, z: Q. 1117,3 69.4 171.3 34.1 " 76.5 111.6.
` D. F. 24 26 26 "24 2Q 2d
p< 0.001 0.001 0.001 0.001 0.001 0.001
T '35 6 -9106 19541 ý8? _A 5013 ; <1
T ; "r; o... rý-ýw, E-" d. 2; ß
110
t
S
Table (3.8)
##** 74 /4TH QUARTER **ý`##
FITTING THE TWO PARAMET ER LOGNORM AL 025. BY DIFF. METHODS
TABL. E O F ESTIMATED PARAMETER S AND STAT ISTICS
METHOD--> MAXIMUM 7(, 93 % 27F', 73 01, 'MOMNTS LIKELIHOOD QUANTILES DUANTILES GRAPHICAL . REGRESSION
STATISTICS -.
MEWHAT 4.757 4.658 4.491 4.702 4.727 4.730
SIGMA2HAT 0.807 1.103 1. "278 1.021 0.836 0.833
ALPHA=MEAN 174.1116- 182.958 168.915 , 183.668 171.620 171.769
BETA=S. D. 194.033 259.553 271.767 244.844 196.111 195.838
COF OF VAR 1.114 1.419. 1.609 1.333 1.143 "1.140 i
MEDIAN 116.361 105.410 89.168. 110.215 113.000 113.26a
MODE 51.927 34.990 24.848 39,087 ..
49.001 49 . 24A
SKEWNESS 4.724 7.111 8.991 6.368 4.921 4.902
KIURTOSIS 59.784 161 , 2.74 293.892 122.451 65: 763 65.176
TOTAL CHISZ" 215.3 61.3 166.1 63.8 160.8 167.6
25 27 27 27 . 25 25
pC 0.001 0.001 0.001_, 0.001 0.001 0.001
T 6102 -15853 32546 -20139 12375 13712
ký " - 1 -3%' 6% -3. aö 2.3;; 2: 6'ý cosT
111
/r
F
Table (3.9) .
,
xý ,. xx 75 /1; 'i (1LfAfi'i Efi ý: * xx
F :[TYx. d; i 'i HE YW(1 P/ fil; r. 1E Y EH L(1GN(1RM Hl. [)I " HY C)FCFF " METHODS
A' CF E:: i'i : ý: "-; r; 'i E[i fýP, R MME'i E R1 1i AND S i, T xs'i IG-i
MET HO[)--> i4AX 3'MUJ'-" 27`*: 13 ý4C1(iIN- ; LXKEl_1H(1('C) : 4tiA; V Ty LE .ý' J! 1. .
Nýiý : L[- E f1 A I. 1TCHL f'HE i I(: id
""""". """"""""""" """""s"""""" """""""""" """""r""" """"""""""""""". ""w
^IEWHAT a""JaF, 4 1661u a"5() el a, 711 4,14'; d, 7(fi!
11: 1f C1; 'i (I " Fi1 z) 1. (l )6 1 "21 R 1 . '(1 12. C) "1 41i (I " R65,
ftl. i*i='kIEA N 173"((! 2 102.4`-, '' 16R"6R(1 1A4. _1R 175 . 6FA R 170,11°
5.3(1 2Ci: i"f fl 265"FO') 2 4.3.961 2(12 . c) 13 1UC'"306
Ct)i aF " V: \ f; 1.12 1 . 30f1' 1. '7C, 1.32'3 1.1 R5 r
1.172
:c (1 (fltiýJ 175.1?. R 9(Ih. 5'11 «(1.392 111.1'7 115"00 0 11f1"/11e1
M (1Cl(: 50 . 750 2ý, "04 14 [i (1 "[110 CIýJ, 273 4ý , 513
c: i . '1ý:: [i C"sýil(ý P ho R, h4(J h. 1Rf) 5; (IOfi ß) .
1?. 7
ý 1ýý: ýi ýw: i l: i ý'ý .J 71 1 CI'J
" i: iýi jrýý-
" (lýav i1 ýý
,!, h, 2 (
.4 L2 72 .
413
TOTAL 149.6 70.9 164.7 72.2 125.1 . 113.7
D. F. 24 27 27 21 244 ;4
P< 0.001. . 0.001' 0.001 0.001 0.001 0.001
T 3370 -14909 24317 21324 -368 14956
0.7 , _ 3.3; - 5.4': -4.7. '')' -0 1 zj
112
A
Table (3010)
* 75 /2ND QUARTER *# *-
FITTING THE TWO PARAMET ER LOGNORMAL"DIS" BY DIFF. METHODS
TABLE OF ESTIMATED PARAMETERS AND STAT ISTICS
' METHOD--> MAXIMUM 7e93 27E73 °ý . MOMNTS LIKELIHOOD QUANTILES QUANTILES GRAPHICAL REGRESSION
STATISTICS
ME1, '+HAT 4.778 4.687 4 "544. 4.720 4.754 4.754
9, IGMA2HAT 0.830 1.103 1.270 1.021 0.82A 0.854
ALPHA=MEAN 180.083 188.317 177.515 186.914 175.482 177.882
BETA=S"D. 204.790 267.145 284.045 249.142 199.193 206.621
COF OF VAR 1.137 1.419 1.600 1.333 1.135 1.162
MEDIAN 118.919 108.501 94.078 112.170 116.000 116.056
MODE 51.857 36.018 26.423 40.397 50.688 49.401
SKEWNESS 4.882 7.111 8.897 (1.367' 4.868 5.052
KURTOSIS 64.551 161.244 285.982 122.392 64.111, 69.946
T0T/L CHISQ" 145.1 50.6 121: 8 52.3 140.2 116.5
D. F. 24 27 27 27 24 27
p 01001 0.004 0.001 0.003 0.00. 0.001
T 4871
T COST t Till' T.
-5951 25634 -10189
ý1. ý`ä 5.?; '4)
15935 11176
3.51 , 2.5%
113
**l Y W0.4)AIIAr�CYFR L0GN0f1MAL [)I 3. x 73/4T; -+ QUARTER DATA
ESTIMATION EBY'MULTINOMIAL MAX. LIKELIHOOD METHOD : - MEW = 4.5163 " SIG%IA2=-. 1.0553
MEAN= 155.080 S. D. = 212.237
N
AMOUNT £ ACTUAL. EXPECTED ACTUAL- EXPECTED CL. NO. Cl.. NO. EXPECTED LOSS (A-E)**2/E
1- 30 47R. + 434. 44. 682. 4.1161 31- 6(}
, 518. 613. -95. -4322. 14.723
6,9C 461. 463. -2. -151. 0.009 91- 120 359. 335. 24. 2532. 1.719
121- 150 239. 245. -6. -813. 0.14i* 7 151- 160 213. 182. 31. 5130. 5.280 181- 210 "140. 139. 9. 1759.
. 0.: )83
211-- 240. ' 102. 107. -5. -1127. 0.234 24 1- 270 81. 84 . -- 3. -767. 0,10 7 271- 300 56. 67. -9, -2509, 11209 301-330 66. 55. 11. 347.0. 2.200
i- 361 45. 45. 0. 0. 0.000 361- 390 39. 37.
. 2. 7` 1. 0.108 391- 420' 35. 31. 4. 1622. " 0.516 421- 450 34, 26. 8. 3484. 2.462 451- 480 20'. 22. -2. -931. 0.182 491- 510 29. 19. 10 , 4955. '5 . 263 511- 540 14. 16. -10 r'
7'1 -
^2 -'; _1", ri n -, c
-651. u. 033 701- 800 '18. 19. -1. -"751. 0.053 601- 900 20. 13. 7. 5953. 3.769 901-1000 . 6. ' 9. -3. -2FI51. 1.000
9001-110: 1 4. 7. -3. -3151. 12E? f, 1101-12(10 4. 5. -1. -1150. 0.290" 1201-1300 1. 4. -3. -3751. 1301-1400 3. o... 0.
1401-1500 1. -1. -145(). 1.778 15; 11-16(10 0. 2. -2. -3101. 16(11-1700 1. 1. (, 0. 1 701-"'1ä0O 0. 1. -1 . -1751 . 19Q1-l900 0. 1, -1. -1650. 1901-2000 1. 1. (l. 0. 2001-2100 0. 1.. -1. -2050. 2101-2200 0. 0. 0, 0', 2201-2300 0. 0., . 0. 0. 23(11-24(1"
------- 1.
----------- (1k
----------- 1.
----------- 23 5(l.
---- - ' P. -ea6
---- TOTAL 3045. 3C; 4` .
-------- - -F,:,: ý "
----- 1----ý 53.375
e. - 25 " P ýý 1 < Q. ü TOTAL EXP, L. )ý; s .
TO AL AT /ý L/ A Y. l ll
5I 1ý. 0*1
pC. 114
Table (3: 12)
T: UO-PARAMETER LOGNORMAL DIS.
74/18T (dUP. f1TER DATA I
ESTIMATION 'Y MULTINOJ"MAL MAX. LIKELIHOOD 'V, ETI! OD : - MEi'1 = 6.5098 SIGP4A2- 1.0572
MEAN^ 1,54.070 S. D. = 211.156
AMOUNT £ ACTUAL EXPECTED ACTUAL- EXPECTED CL.. NO. CL. NO. EXPECTE D LOSS (A-E)**"2/E
1- 30 381. 352. 29. 450. 2.389 31- 60 428. 493. -65. "-2957. 8.570 61= 90 351. 372.
- -21. -1585. 1.185 91- 120 334. 26A. 66. 6963. 16.254
121- 150 , 211. 195. 16. 2168ß 1.313
151- 180 133. 145. -12. -1986. 0.993 181- 210' 98. 110., -12. -2346. - 1.309 ý1 1- 240 82. 115. -3. -677. ' 0.106 241-, 270 54. 67. -13. -3321. 2.522 271- 3(1(1 52. -
54. -2. -571. 0.074 301- 330 53. 43. 1t1. 3155. 2.326 331- 360 36. -35. 1. 346. 0.029 3E61- 390 29. 29. 0. 0. 0.000 391- 420 26.. 24. 2. P11. 0.167 421- 450 22. 20. 2. 1371. 0.200
451- 480 22. 17. 5. 232'7. 1.471 481- 510 17. 15. 2. 9911. 0.267 511- 540 10. 13. -3. -1576. 0.692
541- 570 19. 11. P. 4444. 5.818 571- 600 4. 9. -5. -2927. 7.778 601- 700 26. 24. 2. 1301. 0.167 701- 800 21. 15. 6. 4503. 2.400- Sol- 900 11. 11. 0. 0. 0.000 901--1000 10. 7. 3. 2851. 1.286'
1001-1100 5. 5. 0. 0. 0.000 1101-1200 2. 4. -2. . -2301. 1201-1300 1. 3. -2. -2501. 2.2r<, 1301-1400 2. 2. 0. . D. 1401-1500 11. 2. -2. -2901. 1501-1600 0. 1. -1. -1550. 1601-1700 0. 1. -1. -16511. 1701-1600 is 1" 1)" 0. 2.2s? 6
TOTAL 2441 .' 2433. 2329. -
56.0-9.6 --
. ----------- --------. --- -------- -------- ------------ --------------- O. F. 24
TOTAL EX'r'. LOSS P `0.0c! 0.6 f1, KOL - S1 T, ft? JGV ý00? _; r0 Ti! AC: 7. C['S
P < 0.01
1'5
*#* TWO-PARAMETER LOGNORMAL DIE.
1/2ND [ JA. i1TEH DATA
ESTIMATION BY MULTINOMIAL MAX. LIKEi_IH000 METHOD MEW - 4.5462 SIGMA2= '1. n 126
MEAN= 156 : 407 S. D. = -207.061
AMOUNT £ ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A-E)**2/E
1- 30 -3`51. 312. 39. 60 . 4.875 "31- 60 380. 473. -93. -4231, .
18.205 61- 90 382. 367. 15. 1132. 0; 613 91- ' 120 295. 268. 27. " 2848. 2.720
121- 150 211. 1970 14. 1897. (). 995 151- 180 142. 147. -5. -828. 11.171) 181-" 2101 114. 112. 2. 391, f). 033 6 211- 240- 101. ' - 87. 14. 3157. 2.253
. 241- 270 57. 68. -11. -21110. 1.779 271-. 3n0 51: 54. -3. -857. '0.167 301- 330 39. 44. -5. -157.7. 0.5E, ß; 331- 360 30. 36. . n. ' 0, 0. CIO0 361- 390 25. 29. -. -4. -1502. 0.552
'391- 420 24. 24. 0, - 0. 0.000 421- 450 27.
" 21. 6. 2613. 1.714
451- 480 18. 17, 1. 466. 0. n59 481- 510 21, 15. 6. 2973. 2.400 311- 540 17.. 13. 4. 2102. 1.231 541- 570 12, 11. 1. 555. Ell. 091 571- 600 11. 9. 2. 1171. 0.444 601- 700 30. 23. 7. 4553. 2.130 -701- 800 13. 15. -2. -1501. 0,267 801- 900 11, in. 1. 851. 0.100 901-1000 4. 7. -3. -2851. 1.286
1001-1100- 7. 5. 2. 2101, 0.800 1101-120() 0. 4. -4. -4602, 1201-1300 1" 3. -2. -2501. 5), 142 1301-1400 1. 2. -1. -1350, 1401-º500 0. -2. . -2901, 1501-1600 1. 1. 0. 0. 1601-1700 0, 1. -1. "-1650.
" 1701-1800 0. 1. -1. -1751. 1E? 01-1900 1. 1. 0. 0. 3,12
TO TAL -------
-----------
----------
? 38i. ----------
-
2370,
----------- --------
_ -349Fi.
ý.. -
---- ------------
-Sý. rO;
-----
------------
TOTAL EXP. LOSS P \.. 0.001
------ ---------_ TOTAL ACT. COST KOI, - S: '. I "c7 T 0.0^3
P < 0.01
116
' #* TWO-PARAMETER LOG"yfRMAL DIS. *#*
"74/3RD QUARTER DATA
" ESTIMATION BY MULTINOMIAL MAX. LIKELIH00D METHOD -MEW = 4.6371 SIGMA2= 1.0112
MEAN= 171,1 9 S. D. = 226.355
I% AMOUNT £ ACTUAL EXPECTED ACTUAL- EXPECTED
CL. NO. CL. NO. EXPECTED LOSS (A-E)*"*2/E 1- 30 362. 315. 47. 726. 7.013
31- 60 427. 518. -91. -4140. . 15.986 61- 9T1 383. 421. -38. -2869. 3.430
. 91- 120 356. 317. 39. 4115. 4.798 121- 150 283". 238. 45. 6097, 8.508 151- 180 194. 181. 13. 2151. 0.934 181- 210 137. 140. -3. -587, 0.064 211- 240 97. 110. -13. -2931. 1,536 241- 270 86. 87. -1. -256. 0.011 271- 300' 71. 70. 1. 286, 0,014 301- 330 64. 57. 7. 2208. 0,860 331- 360 45. 47. -2. -691. 0. Q85 361- 390 44" 39. 5. 11377. 0.641 391- 420 25. 33. -8.1 -3244. 1.939 421- 450 26. 27. -1. -436. 0.037 451- 480 22, 23. -1. -466. 0.043 481- 510 25. 20. 5. ' 2477. 1.250 511-'540 14. 17. -3. "-1576. 0,529 541- 570 14. 15. -1, -555. 0.067 571- 600 17. 13. 4. . 2342. 1.231 601- 700 - 32. 32. 0., 0. ' 0.000 701- 800 34. 21. 13. 9756. 8.048. 801- 900 17. 15. 2. 1701. ' 0.267 901-1000 4. 10. -6. -5703. 3.600
1001-1100 9. 7. 2. 2101. 0.571 1101-1200 4. " 5. -1, -1150. ,
0.200 1201-1300 " 0. 4. -4. -5002. 1301-1400 1. 3. -2. -2701.
' 5.115
1401-1500 3. 2. 1. 1450, " . 1501-1600 Q. 2. -2. -3101. 1601-1700 1. 2. -1. . -1650. 0.667 1701-1800 0. 1. -1. -1751. 18o1-1900 0. 1. -1. -1850. 1901-2000 0.. 1, -1. -1950. ' 2001-2100'" 0. 1. -1. -2050, 2101-220.0 1. 1. 0. 0. 2201-2300 0. 0. 0.. 0. 2301-2400 . 0. 0. 0. 0. 2401-2500 0. 0. 0. 0. 2501-2600 1. 0. . 1 1. 2550. 1. COG
: E:::::: : EEEEI::: :: EE::::: ::::::::::::: EEI... '69,27A
1)- . _26
TOTAL EXP. LOSS P<0.001 ---------------- TOTAL ACT. COST X CL - 'D X0.029
i1T P<0.01
* TWO-PARAMETER LOGNORMAL DIS, ##*
74 /4TH QUARTER DATA.
ESTIMATION BY MULTINOMIAL MAX. LIKELIHOOD METHOD i MEW = 4.6722 SIGMA2= 1.0562
UUEAN- 181.329 S. D. = 248.325
'¼
AMOUNT £ ACTUAL 'EXPECTED ACTUAL- EXPECTED.. CL, X00. CL. NO. EXPECTED LOSS (A-E) **2 /E
1-- 30 394. 340. 54. 837. 8.576 31-- 60 452. 547. -95. -4322. 10.499 61- 90 426, 447. -21. -1585. 0.987 91--120 348. 339. 9. 950. 0.239
121- 150 272. 257. 15. 2032. 0.875 151- 180 219. 198. "21.
_ 3475. 2.227 181- 210 154. " 154. 0. 1 0. 0.000 211- 240 124. 122. 2. 451. 0.033 241--270 105. 98. 7. 1788.
"0.500 271- 300 78.. 79. -1. =286. 0.013 301- 330 75. 65. 10. 3155. 1.538 331- 360 58. 54. 4. 1382. 0.296' 361- 390 55. 45. 10. 3755. 2.222 391- 420 29. 38. -9. -3649. 2.132 421- 450 43. 32.
. 11. 4790. 3.781 451- 480 24. '. 28. -4. -1852. 0.571 4111- 510 22. 24. -2. -991. 0: 167 5 540' 24. 20. 4. 2102. 0.800 541- 570 19. 18. 1. 555'. 0.056 571- 600 14. 16. -2. -1171. 0.250 601- 700 42. 39. 3. 1951. 0.231 701- 800 211.. 26. 2. 1501. 0.154 801- 900 20. 18". 2., 1701. 0.222 901-1000 17. 13., 4. 3802. 1.231
1001-1100 8. 10. -2. -2101. 0.400 1.101-1200 5. 7. -2. -2301. 0.571 1201-1300 1. 5. -4. -5002; 3.200 1301-1400 5. 4. 1. 1350. 1401-1.500 0" 3. -3. -4351 . 0.600 1501-1600
.. 1. 3. -2. -3101.
1601-1700 "0. . 2. -2. -3301.
1701-1800 0. '
1 2. -2. -3501. Sýýa3 1801-1900 0b 1. -1. -1850. 1901-2000 0. -1. -1950. 2001-2100 1. 1. . 0. Q, .. 2101-2200 -2150. 2201-2300 0. 1. -1. -2250. 2301-2400 1. " 1. 0. 0. 2.667
--------------, ------------------. -------------------- ----------- TUTAL. ----------
3064. ----------
3059. ------------ -----------
-10147. ------------- ----- -------
D. F. = 27
TOTAL EXP. LOSS F <. C. 001
------- -------- TOTAL ACT. COST ,., Ko'
"F '; C. C2 . 118
Tab. 7f /1 STi QUARTER DATA
ESTIMATION By MULTINOMIAL M. AX. LIKELIHOOD METHOD : MEW - 4,6839 8IGMA2= 1.0241
MEAN= 180.540 S. D. = 241.173
AMOUNT £ ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A"-E) **2 /`
1- 30 324. - 275. 49. ' . 760. 8.731 31- 60
, 31-17. 463. -76. -3458. 12.475
"61- 90 345. 384. -39. -2944. 3.961 91- 120 289. 293. -4. -422. (1,055
121- 150 " 253. 223. 30. 4065. 4.036 151- 180 187. 171. 16. 2648.
. 1,497
181- 210 138. 133. 5. 978. 0.188 211- 240 114. 105. 9. 2029. 0.771 241- 270 93. 84, 9. . 2299. 0.964 271- 300 67. 68, -1. -286. 0.015 301- 330. 63. 56. 7. 2208. 0.875 331- 360 44, 46.. -2. -691. 0.087 361- 390 44. 39. 5. 1877. 0.641 391- 420 " 35. 32. 3. 1216. 0.281 421- 450 25.
. 28. -3. -1306. 0.321
451- 480 26. 23. 3. 1396. . 0.391
481- 510 18. ' 20. -2. -991. 0.200 511- 540 18. " 17. 1, 525. 0.059. 541- 570 22. 15. 7. 3888. 3.267 571- 600 '. 17. 13. 4. 2342. 1.231 601- 700 39.
. 33. 6. 3903. 1.091
701- 600 19. 22. -3. -2251. 0.409 801.900 18. - 15. 3. 2551. 0.600 901-1000 12. 11. 1. 951. 0.091
1001-1100 3. 8. -5. -5252. 3.125 1101=1200 1. 6. -5. -5752. 4.167 1201-1300 11 4" -3. ' -3751. 1301-1400 01 3. -3. -4051. Z 5.14Z 1401-150[) 1. 3. -2. -2901. 1501-1600 0. 2. -2. -3101.
. 1601-1700 1. 2. -1, -1650. z" "571 ý . 1701-1800 0. .1. -1 " -1751. 1801-1911. ) 0. 1. -1� -185,0
, 1901-2000 0+ 1. -1. -1950. 2001-2100 1. 1.4 01 0. 2101-"2200 1.
, 1. 0. 0,
2201-2300 0. _
1. -1. -2250. 2301-2400> 0. 01 01 2401-250'0 0. 0. 0. 01 2501-2600 0. 0. 0. 01 2601-2700 0. a. 01 01 2701-28[)0 0" 0" 0. 0' .e 2801-2900 0. 0. t). 0. 2901-3000 0. 0. 01 0. 3001-3100 0. 0. 0. 0, 3101-3200 0. 0. 09 00 3201-3300 0. 00. a. 0. 3301-34(10 0. 01 0, 0. 3401-3500 0. 0. C� 0 3,01-3600 1. 0. 1. - 3550.
Ti3S AL ------------
2607. ----------
2603. -------------
--__--_-
------
_- -ý- -9423.
--, ý;
_.. _ 50.7-
TO AL EXP LOSS ---- ------------ --------
T
- .
_ _-. - ---_- _ ý -2.. 1% 'Q. 01 p ý'>
... !
r0TAL A,;, T , COST
'Talb? e (31-17)
# TWO-PARAMETER LOGNORMAL DIS .##
75/2NU QUARTER DATA
ESTIMATION BY MULTINOt. 1IAL MAX. LIKELIHOOD METI-10D MEW = 4.7006 SIGMA2= 1.0562
MEAN= 186.551 S. D. = 255.479
AMOUNT F, ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. No. EXPECTED LOSS (A-E )* '2 /E
1- 30 3.02. 264. 38. 589. 5.470 31- 60 374. 435. - -61 . -27751.. 8.554 61- 90 332. 360. -28. -2114. 2.178, 91- 120 277. 276. . 1. 105. . 0.004
121- 150 235. 211. ' 24. 3252. 2.730 151- 180 187. 163. 24. 3972.
. 3.534 181- 210 122. 127.
. -5. -978. 0.197 211- 240 110" . 101. 9. 2029. 0.802 241- 270 80. 81. -1. -256. 0.012 271- 300 72. 66. 6. 1713. 0.545 301- 330 47. 55. -8. -2524. 1.164 331- 360. 39. 45. -6. -2073. 0.800 361- 390 40. 38. 2. 751. 0.105 391- 420 38" 32. 6. 2433. . 1.125 421- 450 29. "27. 2. 871. 0.148 451- 480 21. 23. -2. . -931. 0.174 481 *- 510 30.
. 20. 10. 4955. 5.000
511- 540 19. 17. 2. 1051. 0.235 541- 570 17. 15. 2. 1111. 0.267 571- 600 11. 13. -2. -1171. 0.308 601- 700 - 36.. 34. 2. 1301. ' 0.. 1 1A 701- 800 22. 23. -1. -751. 0.043 801- 900 22. if,, 6. 5103. 2.250 901-1000 111 1 1. () . 0. 0.000
1001-1100 4. 8. -4. -4202. 2.000 1101-1200 3. 6.
' -3. -3451. 1.500'
-1201-1300 3.
. 51 -2. -2501. p. EO0
1301-1400. 6. -
4. ' 2. 201. 1401-1500 " 2. 3. -1 . -1450. 1 1 J3 1501-1600 1. 2. -1. -1550.
.
1601-1700 0., 2. -2. -3301. 1701--1800 1. 1" 0.
. 0" 1.800
1801-1900 0. 1. ' -1. -1850. ' 1901-2000 1. 1. '. 00 " 0. 2001-2100 0. 1. -1. -2050. 2101-2200 0. '1. -1. -2150. 2201-2300
---
1. -----
------
11 -----------
0. ----------
0. ............ -
11000 -
TOTAL
- -----------
2495. " ----------
. 2.489.
----------- -----------
----
-4142'. -----------
-------------
44.605
------------- D.
- F. a 27
TOTAL EXP. LOSS P<0.02
--------------- _ -O. 9 TOTAL ACT. COST YOL -. 3MIP OV D .. 0,020
1'o
Table (3.18)
**' TWO-PARAMETER LOP NORMAL DIS. ***
PREDICTION OF 74/4TH QUARTER CLAIMS COST USING 73/4TH QUJARTER, MULTIN PMIAL MAXLIK. PARAMETERS
MEW= 4.516 SIGMA2_1.056 INFLATION RATE 1= 0.07* C ALCULATED FROM INFLATION IGNORED.
PREDICTION PAR AMETERS ARE : - MEW= 4.516 SIGMA2_1.056 MEAN CLAIM AMOUNT= 155.09 S. D. L212.36 ACTUAL 74/4TH PARAMETE RS : - MEW= 4.672 SIGMA2=1.056 MEAN CLAIM AMOUNT= 181.27 S. D. =248.21
AMOUNT £ AC T. NO. E XP. NO. A-E EXP. LOSS (A-E)**2/E 1- 30 394. 437. -43. -667. 4.231
31- 60 452. 616. -164. -7462. 43.662 61- 90 426. 466. -40. -3020. 3.433 91- 120 34.9. 337. l i. 1160. 0.359
121- 150 272. 246. 26. 3523. 2.748 151- 180 219. 183. 36. 5958. 7.082 181- 210 154. 139. 15. 2933. 1.619 211- 240 124. 108. 16. 3608. 2.370 241- 270 105. 85. 20. 5110. 4.706 271- 300 78. 68. 10. 2855. 1,471 301- 330 75. 55. 20. 6310. 7.273 331- 360 58. 45. 13. 4491. 3.756 361- 390 55. 37. 16. 6759. 8.757 391- 420 29. 31. -2. -811. 0.129 421- 450 43. 26. 17. 7403. 11.115 451- 480 24, c2. 2. 931. 0.182 481- 510 22. 19. 3. 1486. 0.474 511- 540 24. 16. 8. 4204. 4.000 541- 570 19. 14. 5. 2777. 1.786 571- 600 14. 12. 2. 1171. 0.333 601- 700 42. 30. 12. 7806. 4.800 701- 800 28. 20. A. 6004. 3.200 801- 900 20. 13. 7. 5953. 3.769 901- 1000 17. 9. 8. 7604. 7.111
1001-1100 8. 7. 1. 1050. 0.143 1101-1200 5. 5. 0. 0. 0.000 1201-1300 1. 4. -3. -3751. 1301-1400 5. 3. 2. 2701. 1401-1500 0. 2. -2. -2901. 1501-1600 1. 2. -1, -1550. 1601- 1700 0. 1. -1. -1650. 1701-1800 0. 1. -1. -1751. - 1801 1900 0, 1. -1. -1850. 1901-2000 0. 1. - 1. -1950. 2001-2100 1. 1. 0. 0. 2101-2200 0.4 0. 0. 0. 2201-2300 0. 0. 0. 0. 2301-2400 1. 0,
-- - 1.
-------- 2350.
- ------------------------- TOTAL 3064 3062
----------- 66786. ----------
128.509
CHI SQ. STAT. =13?.; , D. F .. ýO P<O. C01
TOTAL ACTUAL COST = 533707. TOTAL EXPECTED COST = 466921.
TOTAL EXP. LOSS
--------------- 12.51 ll, TßTAI_ ACT. COST 12
*** TV, Wun-F'ARnr. Mrc"" GNPiJntvS un.
nLnt_ nZ$. 1aý 1. ie } r_fi
Iý. üuýv %ýKý n` jýýý
PREDICTION OF 75/1ST QUARTER CLAIMS COST USING 74/1ST QUARTER MULTINOMIAL MAXLIK. PARA METERS
MEW- 4.509 SIGMA2=1.057 INFLATION RATE I= 0.0% CALCULATE[) FROM INFLATION IGNORED.
PREDICTION PARAMETERS ARE : - MEW= 4.509 SIGMA2.1.057 MEAN CLAIM AMOUNT= 154.08 S. D. -211. o4 ACTUAL 75/1ST PARAMETE RS : - MEW= 4.684 SIGMA2=1.024 MEAN CLAIM AMOUNT= 180.55 S. D. =241.17
AMOUNT £ ACT. NO. E XP. NO. A-E EXP. LOSS (A_E)**2/E 1- 30 324. 376. -52. -806. 7.191
31- 60 387. 527. -140. -6370. 37.192 61- 90 345. 397. -52. -3926. 6.811 91- 120 289. 286. 3. 316. 0.031
121- 150 253. 209. 44. 5962. 9.263 151- 180 187. 155. 32. 5296. 6.606 181- 210 138. 118. 20. 3910. 3.390 211- 240 114. 91. 23. 5186. 5.813 241- 270 93. 72. 21. 5365. 6.125 271- 300 67. 57. 10. 2855. 1.754 301-'330 63. 46. 17. 5363. 6.283 331- 360 44. 38. 6. 2073. 0.947 361- 390 44. 31. 13. 4881. 5.452 391- 420 35. 26. 9. 3649. 3.115 421- 450 25. 22. 3. 1306. 0.409 451- 480 26. 18. 8. 3724. 3.556 481- 510 18. 16. 2. 991. 0.250 511- 540 18. 13. 5. 2627. 1.923 541- 570 22. 12. 10. 5555. 8.333 571- 600 17. 10. 7. 4098. 4.900 601- 700 39. 25. 14. 9107. 7.840 701- 800 19. 16. 3. 2251. 0.563 801- 900 18. 11. 7. 5953. 4.455 901-1000 12. A. 4. 3802. 2.000
1001- 1100 3. 6. -3. -3151. 1.500 1101- 1200 1. 4. -3. -3451. 1201-1300 1. 3. -2. -2501. 1301-1400 0. 2. -2. -2701. 1401-1500 1. 2. -1. -1450. 150 1-1600 0. 1. -1. -1550. 1601- 1700 1. 1. 0. 0. 1701-1800 0. 1. -1. -1751. 180 1-1900 0. 1. -1, - 1850. 1901-2000 0. 1. - 1. -1950. 200 1-2100 1. 0. 1. 2050. 2101-2200 1. 0. 1. 2150. 2201-2300 0. 0, 0. 0. 2301-2400 0. 0. 0. 0. 2401-2500 0. 0. 0. 0. 2501-2600 0. 0. o. 0, 2601-2700 0. 0. 0. 0. 2701-2800 0. 0. o. 0. 2801-2900 0. 0. 0. 0. 2901-3000 0. 0. 0. 0. 3001-3100 0. 0. 0l 0, 3101-3200 0. 0. 0. 0. 3201-3300 0. 0. 0, 0. 3301-3400 0. 0. 0. 0. 3401-3500 0. 0. o. 0, 3501- 3600
----------- --
1.
---
0. 1. 3550. - - ------------
TOTAL 2 c607 2602 --------- ----------
60567. ---------- 135.703
Table (3.19) - continued
CHI SQ. STAT. =141.1 , D. F. _ 27 P<0.001
TOTAL ACTUAL COST = 452059. TOTAL EXPECTED COST _ 391491.
TOTAL EXP. LOSS
--------------- = 13.40 % TOTAL ACT. COST
0
123
Table (3-20)
*x TWO-PARAMETER LOGNORMAL DIS. ***"
PREDICTION OF 75/2ND QUARTER CLAIMS COST USING 74/2ND QUARTER MULTINOMIAL MAXLIK. PARAMETERS ;
MEW= 4.546 SIGMA2r1.013 INFLATION RATE I= 0.0% CALCULATED FROM ; INFLATION IGNORED.
PREDICTION PARAMETERS ARE ; - MEW= 4.546 SIGMA2-1.013 MEAN CLAIM AMOUNT= 156.41 3. D. =207,14
. ACTUAL 75/2ND PARAMETE RS : - MEW= 4.701 SIGMA2=1.056
MEAN CLAIM AMOUNT= 186.61 S. D. =255.51
AMOUNT £ ACT. NO. E XP. NO. A-E EXP. LOSS (A-E)**2/E 1- 30 302. 327. -25. -388. 1.911
31- 60 374. 496. -122. -5551. 30.008 61- 90 332. 384. -52. -3926. 7.042 91- 120 277. 281. -4. -422. 0.057
121- 150 235. 206. 29. 3929. 4.083 151- 180 187. 154. 33. 5461. 7.071 181- 210 122. 117. 5. 978. 0.214 21 1- 240 110. 91. 19. 4284. 3.967 241- -270 80. 71. 9. 2299. 1.141 271- 300 72. 57. 15. 4282. 3.947 301- 330 47. 46. 1. 316. 0.022 331- 360 39. 37. 2. 691. 01108 361- 390 40. 31. 9. 3379. 2.613 391- 420 38. 26. 12. 4866. 5.538 421- 450 29. 21. 8. 3484. 3.048 451- 480 2.1. 18. 3. 1396. 0.500 481- 510 3o. 15. 15. 7432. 15.000 511- 540 19. 13. 6. 3153. 2.769 541- 570 17. 111 6. 3333. 3.273 571- 600 11. 10. 1. 585. 0.100 601- 700 36. 24. 12. 7806. 6.000 701- 800 22. 16. 6. 4503. 2.250 801- 900 22. 11. 111 9355. 11.000 901- 1000 111 7. 4. 3802. 2.286
1001-1100 4. 5. -1. -1050. 0.200 11(31-1200 3. 4. -1, -1150. 1201-1300 3. 3. 0. 01 1301-1400 6. 2. 4. 5402. 1401- 1500 2. 2. 0. 0. 1501-1600 1. 1. 0. 0. 1601- 1700 0. 1. -1. -1650. 1701-11300 1. 1. 0. 0. 1801- 1900 0. 1. -1. -1850. 1901-2000 1. 11 0. 0, 2001-2100 0. 0. 0. 0. 2101-2200 0. 0. 0. 0. 2201-2300 1.
-- 0.
------ 1. 2250.
--------------------- TOTAL 2495 2491
------------------------------ -------- 67002. -
----------- 114.147
----------
GIII SQ. STAT. = 118.5 " D. F .= 27 P<0,001
TOTAL ACTUAL COST = 44930 8. TOTAL EXPECTE D COST = 38230 6.
TOTAL EXP. LOSS
-w 14.91 124 TOTAL ACT, COST
. &able t5.21y---
-`* x TWO-PARAMETER LOGNORMAL DIS. ###
PREDICTION Ur 74/4TH QUARTER CLAIMS COST USING 73/4TH QUARTER MULTINOMIAL MAXI. TK. PARAMETERS ;
MEW= 4.516 SIGMA2=1.056 INFLATION RATE I=29.2' CALCULATED FROM INDEX OF MOTOR VEHICLES REPAIRS COST.
PREDICTION PARAMETERS ARE :- MEW= 4.772 SIGMA2=1.056 MEAN CLAIM AMOUNT= 200.38 S. D. =274.36 ACTUAL 74/4TH PARAMETERS :- MEW= 4.672 SIGMA2=1.056 MEAN CLAIM AMOUNT= 181.27 S. D. =248.21
t
AMOUNT £ ACT. NO. EXP. NO. A-E EXP. LOSS (A-E)**2/E 1- 30 394. 287. 107. 1658 39.892
31- 60 452. 501. -49. -2229. 4.792 61- 90 426. 430. -4. -302. 0.037 91- 120 348. 337. 11. 1160. 0.359
121-= 150 272. 262. 10. 1355. 0.382 151- 180 219. 205. 14. 2317. 0.956 181- 210 154. . 62. -8. -1564. 0.395 211- 240 124. 130. -6. -1353. 0.277 241- 270 105. 106. -1. -256. 01009 271- . 300 78. 137. -9. -2569. 0.931 301- 330 75. 72. 3. 947. 0.125 331- 360 58. 60. -2. -691. 0.067 361- 390 55. 51. 4. 1502. 0.314 391- 420 29. 43. -14. -5677. 4.558 421- 450 43. 37. 6. 2613. 0.973 451- 480 24. 32. -8. -3724. 2.000 481- 510 22. 27. -5. -2477. 0.926 511- 540 24. 24. 0. 0. 01000 541- 570 19. 21. -2. -1111. 0.190 571- 600 14. 18. -4. -2342. 0.889 601- 700 42. 47. -5. -3252. 0.532 701- 800 28. 32. -4. -3002. 0.500 801- 900 20. 22. -2. -1701. 0.182 901- 1000 17. M. 1. 951. 0.063
1001-1100 8. 12. -4. -4202. 1.333 1101-1200 5. 9. -4. -4602. 1.778 1201- 1300 1. 7. -6. -7503. 5.143 1301- 1400 5. 5. 0. 0. 0.000 1401-1500 0. 4. -4. -5802. 1501- 1600 1. 3. -2. -3101. 1601-1700 0. 3. -3. -4951. 1701- 1800 0. 2. -2. -3501. 1801- 1900 0. 2. -2. -3701. 1901-2000 0. 1. -1. -1950. 2001-2100 1. 1. 0. 0. 2101-2200 0. 1. -1. -2150. 220 1-2300 0. 1. - 1. -2250. 2301-2400 1. 1. 0. 0. 2401-2500 0, 1. - 1. -2450, 2501-2600 0. 1. -1. -2550. --
roTAL -
------------
3064 ---------
3063 ---------- -----
-68464,, - -
-----------
--67.603-y
---------- CHI SQ. ST AT. =53.2 , 0" F"= 31 p , 0000'!
TOTAL ACTUAL COST = 533707. TOTAL EXPECTED COST = 602172.
TOTAL EXP. LOSS
j2- ; {ETA ACT, COST
# '# TWO-PARAMETER LORNOHMAL 013.3t## Table i;; 4
PREDICTION OF 75/1ST QUARTER CLAIMS COST USING 74/ 1ST QUARTER MULTINOMIAL MAXI. IK. PARAMETERS
MEW= 4.509 SIGMA2=1.05'7 INFLATION RATE IL33.9"/o CALCULATED FROM INDEX OF MOTOR VEHICLES REPAIRS COST.
PREDICTION PARAMETERS ARE :- MEW= 4.801 SIGMA2=1.057 MEAN CLAIM AMOUNT. 206.32 S. D. =282.72 ACTUAL 75/1ST PARAMETERS :- MEW= 4.684 SIGMA2=1.024 MEAN CLAIM AMOUNT= 180.55 " S. D. =241.17
AMOUNT £ ACT. No. EXP. NO. A-E EXP. LOSS (A-E)**2/E . 1- 30 324. 233. 91. 1410. 35.541
31- 60 387. 415. -28. -1274. 1.889 61- 90 345. 361. -16. -1208. 0.709 91- 120 289. 286. 3. 316. 0.031
121-. 150 253. 223. 30, 4065. 4.03.5 151- 180 187. 176. 11. 1820, 0.688 181- 210 138. 140. -2. -391. 0.029 211- 240 114. 113. 1. 225. 0.009 241- 270 93. 92. 1. 256. 0.011 271- 300 67. 75. -8. -2284. 0.853 301- 330 63. 63. 0. 0. 0.000 331- 360 44. 53. -9. -3109. 1.528 361- 390 44. 44. 0. 01 0.000 391- 420 35. 38. -3. -1216. 0.237 421- 450 25. 32. -7. -3048. 1.531 451- 480 26. 28. -2. -931. 0.143 481- 510 18. 24. -6. -2973. 1.500 . 511- 540 18. 21. -3. -1576. 0.429 541- 570 22. 18. 4. 2222. 0.889 571- 600 17. 16. 1. 585. 0.063 601- 700 39. 41. -2. -1301. 01098 701- 800 19. 28. -9. -6754. 2.893 801- 900 18. 20. -2. -1701. 0.200 901-1000 12. 14. -2. - 1901. 0.286
1001- 1100 3. 11. -8. -8404. 5.818 1101-1200 1. Be- -7. -8053. 6.125 1201-1300 1. 6. -5. -6252. 4.167 1301-1400 0. 5. -5. -6752. 5.000 1401-1500 1. 4. -3. -4351. 1501- 1600 0. 3. -3. --4651. 1601-1700 1. 2. - 1. -1650. 1701-1800 0. 2. -2. -3501. 1801- 1900 0. 2. -2. -3701. 1901-2000 0. 1. - 1. -1950. 2001-2100 1. 1. 0. 0. 2_101-2200 1. 1. 0. 0. 220 1-2300 0. 1. - 1. -2250. 230 1-2400 0. 1. - 1. -2350. 2401-2500 0. 1,
. - 1. -2450. 2501-2600 0. 0. 0. 0. 2601-2700 0. 0. 0. 0. 2701-2800 0. 0. 0. 0. 2801-2900 0. 0. 0. 0. 2901-3000 0. 0. 0, 0. 3001-3100 0, 0. 0, 0, 3101-3200 0. 0. 0. "3ý 3201-3300 0. 0. 0. 0. 3301-3400 0. 0. 01 0. 3401-3500 0. 0. 0. n 3501-3600 1. 0. 1. 3550.
---------------------- ---------------- ----..... ----... -. .. -------- TOTAL 2607 2603 -7153. °,. 74.701
Table (3.22) - continued
CHI SQ. STAT. -£35.1 , D. F. = 30 P<0.001
TOTAL ACTUAL COST = 452059. TOTAL EXPECTED COST = 523597.
TOTAL EXP. LOSS
--------------- _- 15.82 % TOTAL ACT. COST
127
T'' if (3.23) *-x* TIWWO-PARAMETER LOGNORMAL DIS. ***
PREDICTION OF 75/2ND QUARTER CLAIMS COST USING 74/2ND QUARTER MULTINOMIAL MAXLIK. PARAMETERS ;
MEW= 4.546 SIGMA2=1.013 INFLATION RATE I=33.3% CALCULATED FROM ; INDEX OF MOTOR VEHICLES REPAIRS COST.
PREDICTION PARAMETERS ARE :- MEW= 4.833 SIGMA2=1.013 MEAN CLAIM AMOUNT= 208.50 S. D. =276.12
ACTUAL 75/2ND PARAMETERS :- MEW= 4.701 SIGMA2=1.056 MEAN CLAIM AMOUNT= 186.61 S. D. =255.51
AMOUNT £ ACT. NO. EXP. NO. A-E EXP. LOSS (A-E)**2/E 1- 30 302. 199. 103. 1596. 53.312
31- 60 374. 385. -11. -500. 0.314 61- 90 332. 345. -13. -982. 0.490 91-. 120. 277. 277. 0. 0. 0.000
121- 150 235. 219. 16. 2168. 1.169 151- 1130 187. 173. 14. 2317. 1.133 181- 210 122. 138. -16. -3128. 1.855 211- 240 .
110. 111. -1. -225. 0.009 241- 270 80. 91. - 11. -2810. 1.330 271- 300 72. 75. -3. -857. 0.120 301- 330 47. 62. -15. -4732. 3.629 331- 360 39. 52. -13. -4491. 3.250 361- 390 40. 44. -4. -1502. 0.364 391- 420 38. 37. 1. 406. 0.027 421- 450 29. 32. -3. -1306. 0.281 451- 480 21. 27. -6. -2793. 1: 333
. 481- 510 30. 24. 6. 2973. 1.500 51 1- 540 19. 21. -2. -1051. 0.190
54 1- 570 17. 18. -1. -555. 0.056 571- 600 11. 16. -5. -2927. 1.562 601- 700 36. 40. -4. -2602. 0.400 701- 800 22. 27. -5. -3752. 0.926 801- 900 22. 19. 3. 2551. 0.474 901- 1000 11. 14. - 3. -2851. 0.643
1001-1100 4. 10. -6. -6303. 3.600 1101-1200 3. 8. -5. -5752. 3.125 1201-1300 3. 6. -3. -3751. 1.500 1301- 1400 6. 5. 1. 1350. 0.200 1401-1500 2. 4. -2. -2901. 1501-1600 1. 3. -2. -3101. 1601-1700 0. 2. -2. . -3301. 1701-1800 1. 2. -1. -1751. 1801-1900 0. 2. -2. -3701. 1901-2000 1. 1. 0. 0. 2001-2100 0. 1. - 1. -2050. 2101-2200 0. 1. -1. -2150. - 2201-2300 1. 1. 0. 0.
230 1-2400 0. 1. -1. -2350. 240 1-2500 0. 1. - 1.
----- -2450.
-------- -----------
TOTAL -----------
---------
2495 ---------
----------
2494 ----------
-
------
--
"-63269. ----------
------------
82.792 -----------
CHI SO. STAT. =91.9 , D. F. = 30 P <o. ooi
TO TAL TOTAL
TOTAL
TOTAL
ACTUAL COST = 44930A. EXPECTED COST = 512577.
EXP. LOSS
- --14.0A % 128 ACT. COST
Table {. 2d)
## TWO-PARAMETER LOGNORMAL DIS. ###
129
PREDICTION OF 74/4TH QUARTER CLAIMS COST USING 73/4TH QUARTER MULTINOMIAL MAXLIK. PARAMETERS
MEW= 4.516 S; GMA2 =1.056 INFLATION RATE I= 23.5% CALCULATE D FROM INDEX OF'AVE. EAR NINGS, MISCELLA NEOUS SERVICES.
PREDICTION PARAMETERS ARE : - MEW= 4.727 SIGMA2=1.056 MEAN CLAIM AMOUNT= 191.54 S. D. =262.26 ACTUAL 74/4TH PARAMET ERS : - MEV9= 4.672 SIGMA2=1.056 MEAN CLAIM AMOUNT= 181.27 S. D. =248.21
AMOUNT £ AC T. NO. EXP. NO. A-E EXP. LOSS (A-E)**'2/E 1- 30 394. 310. 84. 1302. 22.761
31- 60 452. 522. -70. -3185. 9.387 61- 90 426. 438. -12. -906. 0.329 91- 120 348., 338. 10. 1055. 0.296
121- 150 272. 260. 12. 1626. 0.554 151- 180 219. 202. 17. 2813. 1.431 181- 210 154. 159. -5. -978. 0.157 211- 240 124. 127. -3. -677. 0.071 241- 270 105. 102. 3. 767. 0.088 271-"300 78. 83. -5. -1427. 0.301 301- 330 75. 69. 6. 1893. 0.522 331- 360 58. 57. 1. 346. 01018 361- 390 55. 48. 7. 2628. 1.021 391- 420 29. 41. -12. -4866. 3.512 421- 450 43. 35. 8. 3484. 1.829 451- 480 24. 30. -6. -2793. 1.200 481- 510 22. 26. -4. -1982. 0.615 511- 540 24. 22. 2. 1051. 0.182 541- 570 19. 19. 0. 0. 0.000 571- 600 14. 17. -3. -1756. 0.529 601- 700 42. 43. - 1. -651. 0.023 701- 800 28. 29. - 1. -751. 0.034 801- 900 20. 20. 0. 0. 0.000 901- 1000 17. 15. 2. 1901. 0.267
1001-1100 8. 11. -3. -3151. 0.818 1 10 1- 1200 5. 8. -3. -3451. 1.125 1201- 1300 1. 6. -5. -6252. 4.167 1301- 1400 5. 5. 0. 0. 0.000 1401-1500 0. 4. -4. -5802. 1501-1600 1. 3. -2. -3101. 1601- 1700 0. 2. -2. -3301. 1701- 1800 0. 2. -2. -3501. 1801- 1900 0. 2. -2. -3701. 1901-2000 0. 1. -1. -1950. 200 1-2100 1. 1. 0. 0. 21 01-2200 0. 1. -1. -2150. 2201-2300 0. 1. -1. -2250. 2301-2400 1. 1. 0. 0. 2401-2500 0. 1. . -1. -2450.
- TOTAL 3064
----------------------
3061
------- .
-------- -42168. ----------
51,237 - ---------
CHI SQ. STAT. =65.0 , D. F. = 30 P<o. 001
TOTAL ACTUAL COST = 533707. TOTAL EXPECTED COST - 575876.
TOTAL EXP. LOSS -- -----------= -7.90 %
TOTAL ACT. COST
Týri0 -FAR ; METER LOGNORMAL DIS. Table (3.25)
PREDICTION OF 75/ 1ST QUARTER CLAIMS COST USING 74/1ST QUARTER MULTINOMIAL MAXLIK. PARAMETERS
MEW= 4.509 SIGMA2=1.057 INFLATION RATE I=28.3"/0 CALCULATED FROM INDEX OF AVE. EARNINGS, MISCELLANEOUS SERVICES.
PREDICTION PARAMETERS ARE :- MEW= 4.758 MEAN CLAIM AMOUNT= 197.69
ACTUAL 75/ 1ST PARAMETERS :- MEW= 4.6134 MEAN CLAIM AMOUNT= 180.55
AMOUNT £ 1- 30
31- 60 61- 90 91- 120
121- 150 151- 180 181- 210 211- 240 241- 270 271- 300 301- 330 331- 360 361- 390 391- 420 421- 450 451- 480 481- 510 511- 540 541- 570 571- 600 601- 700 701- 800 801- 900 901- 1000
1001- 1100 1101-1200 1201-1300 130 1- 1400 1401-1500 1501- 1600 1601-1700 1701-1800 1801-1900 1901-2000 2001-2100 2101-2200 2201-2300 2301-2400 2401-2500 2501-2600 2601-2700 2701-2800 2801-2900 2901-3000 3001-3100 3101-3200 3201-3300 3301-3400 3401-3500 3501-3600
SIGMA2.1.057 S. D. =270.90
SIGMA2=1.024 S. D. =241.17
(A-E)**2/E 21.231
4.687 1.437 0.014 4.329 0.971 0.007 0.145 0.180 0.493 0.150 0.720 0.095 0.028 1.161 0.000 1.087 0.200 1.471 0.267 0.000 1.885
0.000 0.077 4.900 5.143 4.167
2687 2611; ----------------------------------------------------------
TOTAL 5O77C+. 54.845
ACT. NO. 324. 387. 345. 289. 253. 187. 138. 114.
93. 67. 63. 44. 44. 35. 25. 26. 18. 18. 22. 17. 39. 19. 18. 12.
3. 1. 1. 0. 1. 0. 1. 0. 0. 0. 1. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0, 0. 0. 0. 1.
EXP. NO. 251. 432. 368. 287. 222. 174. 137. 110.
69. 73. 60. 50. 42. 36. 31. 26. 23. 20. 17. 15. 39. 26. 18. 13. 10. 7. 6. 4. 3. 3. 2. 2. 1. 1. 1. 1. 1. 1. 1.
_ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
A-E 73.
-45. -23.
2. 31. 13.
1. 4. 4.
-6. 3.
-6. 2.
-1. -6.
0.
-5. -2.
5. 2. 0.
-7. 0.
-1. -7. -6. -5. -4. -2. -3. -1. -2. -1. -1.
0. 0.
-1. -1. -1.
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1.
EXP. LO SS 1131.
-2047. -1736.
211. 4200. 2151.
195. 902.
1022.
-1713. 947.
-2073. 751.
-406. -2613.
0. -2477. -1051.
2777. 1171.
0.
-5253. 0.
-951. -7353. -6903. -6252. -5402. -2901. -4651. -1650, -3501. -1850. -1950.
0. 0.
-2250. -2350. -2450.
0. 0, 0. 0, 0. 0, 0. 0, 0. 0.
3550.
Table (3.25) - continued
CHI SQ. STAT. = 68.0 . D. F. = 30 P <0.001
TOTAL ACTUAL COST = 452059. TOTAL EXPECTED COST = 502837.
TOTAL EXP. LOSS
---------------=-11.23 % TOTAL ACT. COST
0
131
Table (3.26)
*" TWO-PARAMETER LOGNORMAL DIS.
PREDICTION OF 75/2ND QUARTER CLAIMS COST USING 74/2ND QUARTER MULTINOMIAL MAXLIK. PARAMETERS
MEW= 4.546 SIGMA2=1.013 INFLATION RATE I=29.5% CALCULATED FROM INDEX OF AVE. EARNINGS, MISCELLANEOUS SERVICES.
PREDICTION PARAMETERS ARE :- MEW= 4.805 SIGMA2a1.013 MEAN CLAIM AMOUNT= 202.55 S. D. =268.25 ACTUAL 75/2ND PARAMETERS :- MEW= 4.701 SIGMA2=1.056 MEAN -CLAIM AMOUNT= 186.61 S. D. =255.51
AMOUNT C ACT. NO. EXP. NO. A-E EXP. LOSS (A-E)**2/E 1- 30 302. 210. 92. 1426. 40.305
31- 60 374. 396. -22. -1001. 1.222 61- 90 332. 350. -18. -1359. 0.926 91- 120 277. 279. -2. -211. 0.014
121- 150 235. 218. 17. 2304. 1.326 151- 180 1887. 172. 15. 2482. 1.308 181- 210 122. 136. -14. -2737. 1.441 211- 240 110. 109. 1. 225, 0.009 241- 270 80. 89. -9. -2299. 0.910 271- 300 72. 73. -1, -286. 0.014 301- 330 47. 60. -13. -4101. 2.817 331- 360 39. 50. -11. -3800. 2,420 361- 390 40. 42. -2. -751. 0.095 391- 420 38. 36. 2. 811. 0.111 421- 450 29. 31. -2. -871. 0,129 451- 480 21. 26. -5. -2327. 0.962 481- 510 30. 23. 7. 3468. 2.130 51 1- 540 19. 20. -1. -525. 0.050 541- 570 17. 17. 0. 0, 0.000 571- 600 11. 15. -4. -2342. 1.067 601- 700 36. 39. -3. -1951. 0.231 701- 800 22. 26. -4. -3002. 0.615 801- 900 22. 18. 4. 3402. 0.889 901- 1000 11. 13. -2. -1901. 0o308
1001-1100 4. 10. -6. -6303. 3.600 1101-1200 3. 7. -4. -4602. 2.286 1201-1300 3. 5. -2. -2501. 0.800 1301-1400 6. 4. 2. 2701.
1401-1500 2. 3. -1. -1450. 1501-1600 1. 3. -2. -3101. 1601-1700 0. 2. -2. -3301. 1701-1800 11 2. " -1. -1751. 180 1-1900 0. 1, -1. - 1850. 1901-2000 1. 1, 0. 0, 2001-2100 0. 11 -1, -2050. 210 1-2200 0. 1. -1. -2150. 2201-2300 11 1. 0. 0, 230 1-2400
-
0.
-------
1, " --------
- 1,
- -2350.
---- ------- TOTAL
------------
-- 2495
---------
2490
--------
------
-------
----------- -44057.
-----------
---------- 65.984
----------
CHI SQ. ET AT. = 72.6 , D. F . -JO P<t'). 001
TOTAL ACTUAL COST = 44930A. TOTAL EXPECTED COST = 403365.
TOTAL EXP. LOSS --------------- _-9.81 % 132 TOTAL ACT. COST
Table (3.27)
*** Two-rAnAMETER LOGNOfl MAL. DIS. ***
PREDICTION OF 74/4TH QUARTER CLAIMS COST USING 73/4TH QUARTER MULTINOf4IAL h"AXLIK. PARA METERS
MEW= 4.516 SIGMA2=1.. 056 INFLATION RATE I= 18.2% CALCULATED FROM GENERAL IN DEX OF RETAIL PRICES.
PREDICTION PAR AMETERS ARE : - MEW= 4.683 SIGMA2=1.056 MEAN CLAIM AMOUNT= 183.32 S. D. =251.00
"ACTUAL 74/4TH PARAMET ERS : - MEW= 4.672 SIGMA2=1.056 MEAN CLAIM AMOUNT= 181.27 S. D. =248.21
AMOUNT £ AC T. NO. EXP. NO. A-E EXP. LOSS (A-E)**"2/E 1- 30 394. 334. 60. 930. 10.778
31- 60 452. 542. -90. -4095. 14.945 61- 90 426. 445. -19. -1434. 0.811 91- 120 348. 339. 9. 950. 0.239
121- 150 272. 258. 14. 1897. 0.760 151- 180 219. 199. 20. 3310. 2.010 181- 210 154. 155. -1. -195. 0.006 21 1-
. 240 124. 123. 11 225. 0.008
241- 270 105. 99. 6. 1533. 0.364 271- 300 78. 80. -2. -571. 0.050 301- 330 75. 66. 9. 2839. 1.227 331- 360 58. 55. 3. 1036. 0.164 361- 390 55. 46. 9. 3379. 1.761 391- 420 29. 39. -10. -4055. 2.564 421- 450 43. 33. 10. 4355. 3.030 45 1- 480 24. 28. -4. -1862. 0.571 481- 510 22. 24. -2. -991. 0.167 511- 540 24. 21. 3. 1576. 0.429 541- 570 19. 18. 1. 555. 0.056 571- 600 14. 16. -2. -1171. 0.250 601- 700 42. 40. 2. 1301. 0.100
701- 800 28. 27. 1. 751. 0.037 801- 900 20. 19. 1. 851. 0.053
901- 100 0 17. 13. 4. 3802. 1.231 1()01-1100 8. 10.. -2. -2101. 0.400
1101-1200 5. 7. -2. -2301. 0.571 120 1-1300 1. 6. -5. -6252. 4.167 1301-1400 5. 4. 1. 1350.
1401-1500 0. 3. -3. -4351. 150 1- 1600 1. 3. -2. -3101. 1601- 1700 0. 2. -2. -3301. 1701-1800 0. 2. -2. -3501. 1801- 1900 0. 1. -1. - 1850. 190 1-2000 0. 1. - 1. -1950. 2001-2100 1. 1. 0. 0.
2101-2200 0. 1. -1. -2150. 220 1-2300 0. 1. - 1. -2250. 2301-2400 1.
- -
1.
------ ------
0.
------- ----- =-- 0.
--------- - ------------------- - ------------------- TOTAL 3064
---------------------
- - 3062
------- --------
- "-16844.
----------
--- -------- 46.748
-----------
CHI SQ. STAT. = 55.5 , D. F.:; Q
TOTAL ACTUAL COST = 533707. TOTAL EXPECTED COST = 550551.
TOTAL EXP. LOSS
"f0". AL ACT. COST
P<O. 003
J\CL - Jig: ;. +1NC D=0.01
9
pý: 0.18
Table (3.28)
*** TýýO-PARAVETEFi LOGNORMAL DIS. *-. t
PREDICTION OF 75/1ST QUARTER CLAIMS COST USING 74/1ST QUARTER MULTINOMIAL MAXLIK. PARAMETERS
MEW= 4.509 SIGMA2=1.057 INFLATION RATE I= 20.3% CALCULATED FROM ; GENERAL INDEX OF RETAIL PRICES.
PREDICTION PARAMETERS ARE - MEW- 4.694 SIGMA2-1.057 MEAN CLAIM AMOUNT= 185.36 S. D. =254.00 ACTUAL 75/ 1ST PARAMETERS : - MEW= 4.684 SIGMA2-1.024 MEAN CLAIM AMOUNT= 180.55 S, D, =241.17
AMOUNT C ACT. NO. EXP. NO. A-E EXP. LOSS (A_E)**2/E 1- 30 324. 280. 44. 682. 6.914
31- 60 397. 457. -70. -3185. 10.722 ° 61- 90 345. 377. -32. -2416, 2.716
91- 120 289. 288. 1. 105, 0.003 121- 150 253. 220. 33. 4471. 4.950 15 1- 180 187. 170. 17. 2813. 1.700 181- 210 138. 133. 5. 978. 0.188 21 1- 240 114. 105. 9. 2029. 0.771 241- 270 93- 85. 8. 2044. 0.753 271- 300 67. 69. -2. -571. 0.058 301- 330 63. 57. 6. 1893. 0.632 331- 360 44. 47. -3. -1036. 0.191 361- 390 44. 39. 5. 1877. 0.641 391- 420 35. 33. 2. all. 0.121 421- 450 25. 28. -3. -1306. 0.321 451- 480 26. 24. 2. 931. 0.167 481- 510 18. 21. -3. -1486. 0.429 511- 540 18. in. 0. 0. 0.000 541- 570 22. 16. 6. 3333. 2.250 571- 600 17. 14. 3. 1756. 0.643 601- 700 39. 35. 4. 2602. 0.457 701- 800 19. 23. -4. -3002. 0.696 Sol- 900 18. 16. 2. 1701. 0.250
41 90 1- 1000 12. 12, 0. 0. 0.000
1001- 1100 3. 9. -6. -6303. 4.000 1101-1200 1. 6. -5. -5752. 4.167 1201- 1300 1. 5. -4. -5002, 3.200 130 1- 1400 0. 4. -4. -5402. 140 1- 1500 1. 3. -2. -2901.
% 150 1-1600 0. 2. -2. -3101. 160 1-1700 1. 2. -1, -1650. 170 1- 1800 0. 2. -2. -3501 . 180 1-1900 0. 1. - 1. -1850. 190 1-2000 0. 1. - 1. -1950. 2001-2100 1. 1. 0. 0, ' 2101-2200 1, 1. 0. 0. 2201-2300 0. 1, - - 1. -2250, 230 1-2400 0. 11 -1, -2350. 2401-2500 0. 0. o. 0 2501-2600 0. 00. 0. . 0. 2601-2700 0. 0. 0. 0. 2701-2800 0. 01 0, 0. 2801-2900 0. 0.. 0. 0. 2901-3000 0. 0. 0. 0. 3001-3100 0. 0, o, 00 3101-3200 0. 0. 0. 0 3201-3300 0. 0, 0. . 0. 3301-3400 0. 0. 0. 0. 3401-3500 0. 0, 0, 0, 350j-360f) I. 0. 1. 3550.
Table (3.28) - continued
---------------------------------------------------------
TOTAL 2607 2606 -23439.46.941 ---------------------------------------------------------
CHI SQ. STAT. = 57.7 ., D. F. = 29 P<o. 003
TOTAL ACTUAL COST = 452059. TOTAL EXPECTED COST = 475498.
TOTAL EXP. LOSS KOT - S: 1'IIRL; OV D= 0.022 --------------- - -5.19 % TOTAL. ACT. COST P=0.15
135
Table (3.29)
* TWO-PARAMETER LOGNORMAL DIS. ý***
PREDICTION OF 75/2ND QUARTER CLAIMS COST USING 74/2ND QUARTER MULTI NOMIAL MAXLIK. PARAMETERS
MEW= 4.546 SIGMA2=1.013 INFLATION RATE I= 24.3^ CALCULATED FROM GENERAL IN DEX OF RETAIL PRICES.
PREDICTION PAR AMETERS ARE ;- MEW= 4.764 SIGMA2=1.013 MEAN CLAIM AMOUNT= 194.42 S. D. -257.48
ACTUAL 75/2ND PARAMET ERS :- MEW= 4.701 SIGMA2=1.056 MEAN CLAIM AMOUNT= 186.61 S. D. =: 255.51
AMOUNT £ AC T. NO. EXP. NO . A-E EXP. LOSS (A-E)**2/E 1- 30 302. 226. 76. 1178. 25.558
31- 60 374. 412. -38. -1729. 3.505 61- 90 332. 357. -25. -1887. 1.751 91- 120 277. 280. -3. -316. 0.032
121- 150 235. 217. 18. 2439. 1.493 151- 180 187. 170. 17. 2813. 1.700 181- 210 122. 134. -12. -2346. 1.075 211- 240 110. 107. 3. 677. 0.084 241-
. 270 f30. 86. -6. -1533. 0.419
271- 300 72. 70. 2. 571. 0.057 301- 330 47. 58. -11. -3470. 2.086 331- 360 39. 48. -9. -3109. 1.687 361- 390 40. 41. - V. -376. 0.024 391- 420 38. 34. 4. 1622. 0.471 421- 450 29. 29. 0. 0. 0.000 451- 480 21. 25. -4. -18626 0.640
-481- 510 30. 21. 9. 4459. 3.857 51 1- 540 19. 19. 0. 0. 0.000 541- 570 17. 16. 1. 555. 0.063 571- 600 11. 14. -3. -1756. 0.643 601- 700 36. 36. 0. 0. 0.000 70 1- 800 22. 24. -2. -1501. 0.167 801- 900 22. W. 5. 4252. 1.471 901-1000 11. 12. -1. -951. 0.083
1001-1100 4. 9. -5. -5252. 2.778 110 1-1200 3. 7. -4. -4602. 2.286 1201-1300 3. 5. -2. -2501. 0.800 1301-1400 6. 4. 2. 2701. 1401-1500 2. 3. -1. -1450. 1501- 1600 1. 2. -1. -1550. 1601-1700 0. 2. -2. -3301. 170 1- 1800 1. 2. -1. -1751. 1801-1900 0. 1. " -1. -1850. 1901-2000 1. 1. 0. 0. 20 0 1-2100 0. 1. -1. -2050. 210 1-2200 0. 1. - 1. -2150. 2201-2300 1. 1. 0. 0.
---------------------- TOTAL 2495
----------------------
------ 2492 ------
--------
--------
----------- -26028.
-----------
---------- 52.728
----------
CHI SQ. STAT. =57.3 , D. F., -- 29 P<0.003
TOTAL ACTUAL COST - 44930R. TOTAL EXPECTED COST = 475336.
TOTAL EXP. LOSS
---. ----------_ - -. 5.79 'Y% TOTAL ACT. COST
<OL-sMPNOý! D=0.030
P -- 0.02
Table (3.: 0)
LOGNORMAL DISTRIBT. JTIOr: **
ESTIMATTY03 PAA-. APO, TER C BY THJ LEAST S rT ARE3 REGRESSION; I'JIOD
7314TH QUAHTM DATA
INPUT
c ]TIIBT
U. o C= 0.00 SSD=0.15923_
5.
C= 5.00 SSD=0.10475_
10. [1
15.0
20. ()
25. n
C= 10.00 SSO=0.0722E1_
C 15.00 SSD=0.05536_
C= 20.00 SSD=0.04983_
G- 25.00 SSD=n, o5281_
18. C)
C= 18.00 6Sa=0.050BA-
19.0
G= 19.00 Be Cl=0.05018__
21.0
Ov 21.00 ")SD-0,04982-
22.
C= 22.00 SSD=0.05013_
137
Table (3.31)
73/4TH (QUARTER DATA
C=01.00 PARAMETERS OF THE LOGNORMAL DIS. ESTIMATED OYL. S. REGRESSION : -
MEW = 4.778 SIGMA SO. = 0.730
MEAN= 150.219 S. 0. = 177.489
SSD=0.050
AMOUNT £ ACTUAL. EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A-E)**2/E
1- 30 478. 499, -21. -325. 0.834 31- 60 518. 504. 14.. 637. 0,389 61- 90 461. 429. 32. 2416. 2.387 91- 120 359, 337. 22. 2321. 1.436
121- 150 239. 260. -21.. -2845. 1.696 151- 180 213. 200. 13. 2151. 0.845 181- 210 148, 154, -6.. -1173. 0.234 211- 240 102. 121. -19. -4284. 2.983 241- 270 81, 95. - -14. -3577. 2.063 271- 300 58. 75.. -17.. -4853. 3,853 301- 330 66, 61. 5. 1577. 0.410 331- 360 45. 49. -4. -1382. 0.327 361- 390 39,. 40, -1, -376, 0.025. 391- 420 35.. 33. 2, all. 0.121 421*- 450 34. 27. 7. 3048. 1.815 451- 480 20. 23. -3. -1396. 0.391 . 481- 510 29, 19, 10, 4955. 5.263 511- 540 14, 16, -2. -1051. 0.250 541- 570 8. 13, -5. -2777. 1.923 571- 600 91 11, -2. -1171. 0.364 601- 700 29. 27. 2. 1301. 0.148 701- 800 18. 17. 1, 751. 0.059 801- 9(10 20, 11. 9. 7654. 7. °364 901-1000 6. 7. -1. -951. 0.143
1001-1100 4. 5. -1. -1050. 0,2C0 1101-1200 4. 3, 1. 1150. 1201-1300 1. 2. -1. -1250. 0.000 1301-1400 3. 2. 11 1350. 1401-1500 1. 1. 0. 0,
Al 1501-1600 0. 1, -1. -1550. 1601-1700 1. 1. 0. 0. 1701-1800 0. 1. -1. -1751. 11301-1900 0. 0. 0. 1901-2000 1. 0. 1. 1950. 2001-2100 0. 0. 0. 0. 2101-2200 0. 0. 0. 0. 2201-2300 0, 0, 0, 0. 2301-2400 1. 0. 1. 2350. ---------- ------------ ---------- ------------ -------- - ------ fl., 142
TOTAL ----------
3045. ------------
3044. ---------- -----------
2660. ----------
35. o/ý0
D. F. X23 TOTAL EXP. LOSS P_ 004
% ---------------= 0.6 % TOTAL ACT. COST
138
Table (3.32)
C=23.00 PARAMETERS OF THE
MEW = 4.819
MEAN= 150.820
SSD=D . 223
74/iST (QUARTER DATA
LOGNORMAL DIS. ESTIMATED ßY L. S. REGRESSION : - SIGMA S(. = 0.679
S. D. = 171.348
AMOUNT r ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A-E)**2/E
1- 30 . 381. 377. 4. 62. 0.042 31- 60 428. 396. 32. 1456. 2.586 61- 90 351. 346, 5. 378.. 0.072 91- 120 334. 276. 58. 6119. 12.188
121- 150 211. 214. -3. -406. 0.042 151- 180 133. 166. -33. -5461. 6.560 181- 210 98. 128. -30. -5865. 7.031 211- 240 82, 100, -18. -4059. 3.240 241- 270 54. 79. -25. -63E17. 7.911 271- 300 52. ' 62. -10. -2055. 1.613 301- 330 53. 50, 3. 947. 0.180
"331- 360 36. 40. -4. -1382. 0.400 361- 390 29. 33, -4. -1502. 0.485 391- 420 26. 27. -1. -406. 0.037 421- 450 22. 22. 0. 0. 0.000 451- 480 22. 18, 4. 1862. 0.889 481- 510 17. 15. 2. 991. 0.267 511- 540 10. 13, -3. -1576. 0. "692 541- 570 19. 111 8. 4444. 5.818 571- 600 4. 9. -5. -2927. 2.778 601- 700 26. 21, 5. 3252. 1.190 701- 800 21. 13. 8. 6004. 4.923 801- 900 11. 8. 3. 2551. 1.125 901-1000 10. 51 5. 4752. 5.000
1001-1100 5. 4. 1. 1050. 1101-1200 2. 2,. 0. 0'167 1201-1300 1. 2. -1. -1250. 1301-1400 2, 1, 1351). 1401-1500 0, 1. -1. -1450. 1501-1600 0, 1, -1. -1550. 1601-1700 0. 0. " 0. 0. 1701-1800 1. 0. 1.
- 1751.
--------- .? 00 .... -----0
TOTAL 2441. 2440.
----- ------- -109.
--- 65.438
------ ----------- U. F. e TOTAL EXP. LOSS
----- ---- ---- P 0.001 TOTAL ACT. COST
139
hk. ---
Table (3.33)
74/2ST QUARTER DATA
C=21.00 PARAMETERS OF THE LOGNORMAL DIS. ESTIMATED BY L. S. REGRESSION : -
MEW = 4.824 SIGMA S(.. = 0.666
MEAN= 152.721 S. O. = 169.077
SSD C. 133
AMOUNT £ ACTUAL. EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS, (A-E)**2/E
1- 30 351. 333, 18, 279.. 0,973 31- 60 -380, 386. -6,. -273. 0.093 61- 90 382. 344, 3A, 2869, 4,198 91- 120 295, 276. 19, 2004, 1,308
121- 150 211, 215. -4, -542, 0.074 151- 180 142, E 166, -24. -3972, 3.470 181- 210 114. 129, -15. -2933. 1.744 211- 240 101, 100,. 1, 225. 0.010 241- 270 57. 79. -22. -5621. 6.127 271- 300 51, 62, -11, -3140, 1.952 301- 330 39, 50. -11, -3470. 2.420 331- 360 36.. 40. -4. -1382. 0.400
"361- 390 25. 32. -7,. -2628. 1.531 391- 420 24, 26, -2. -811. 0.154 421- 450 27, 22, 5. 2177. 1.136 451- 480 18, 18. 0, o. 0.000 481- 510 21, 15. 6. 2973. 2.400 511- 540 17. 12. 5. 2627. 2.083 541- 570 12. 10. 2. 1111. 0.400 571- 600 11, 9. 2. 1171. 0.444 601- 700 30. 21. 9. 5854. 3.857 701- 000 13. 13. 0. 0. 0.000 001- 900 11, 8. 3. 2551. 1.125 901-1000 4. 5. -1, -951. 0.200
1001-1100 7. 3. 4. 4202. 1101-1200 0. 2. -2. -2301. 01800 1201-1300 1. 2. -1. -1250. 1301-1400 1, 1. 0, 0. 1401-1500 0. 1. -1. -1450. 1501-1600 1. 11 01 00 1601-1700 0, 0. 0, 1701-1800 0, 0. 0. 1801-1900 1. 0. 1. 1850. 0.260
TOTAL
---------- 2383.
----------- 2381.
------------
----
------------
---- -829.
---------
-" --- - 100
37
----------- D. F. x 22
TOTAL EXP, LOSS p: 0.02 ---------------= -0.2 % TOTAL ACT. COST
140
Table (3.34)
74/3RD QUARTER DATA
C=19.00 PARAMETERS OF THE LOGNORMAL DIS. ESTIMATED BY L. S. REGRESSION
MEW, = 4.857 SIGMA SO. = 0.731
MEAN= 166.520 S. D. = 192.635
SSD=0.103
AMOUNT £ ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A-E)**2/E
1- 30 362. 369. -7, -108, 0.133 31- 60 427. 433. -6. -273. 0.083 61- 90 383. 388, -5. -378. 0.064 91- 120 356. 315. 41. 4325. 5 . 337
121- 150 283. 248. 35. 4742. . 4.940 151- 18U 194. 195. -1. -165. 0.005 181- 210 137. 153. -16. -3128. 1.673 211- 240 97. 121. -24. -5412. 4.760 241- 270 86p. 97, -11.. -2810. 19247 271- 300 71. 78. -7. -1998. 0.628 301- 330 64. 63. 1. 316. 0.016 331- 360 45, 51. -6. -2073. 0.706 361- 390 44.. 42. 2. 751. 0.095 391- 420 25. 35. -10. -4055. 2.857 421- 450 26. 29. -3. -1306. 0.310 451- 480 22. 24. -2. -931. 0.167 481- 510 25. 21. 4. 1982. 0.762 511- 540 14. 17. -3. -1576, 0.529 541- 570 14. 15. -1. -555. 0.067 571- 600 17, 13. 4. 2342. 1.231 601- 700 32. 31. 1. 651. 0.032 701- 800 34, 19. 15. 11257. 11.842 801- 900 17. 12. 5. 4252. 2.083 901-1000 4. 80 -4. -3802. 2.000
1001-1100 9. 6. 3. 3151. 1.500 1101-1200 4. 4,. 0. 0. 0.000 1201-1300 01 3. -3. -3751. 1301-1400 1, 2. -1. -1350. 3.200 1401-1500 3. 2. 1. 1450,. 1501-1600 0. 11 -1. -1550. 1601-1700 1. 1.. 0, 0. 1701-1800 0. 1. -1. -1751. 1801-1900 01 1. -1. - 1850. 1901-2000 0. 0, 0. 0. 2001-2100 0. 0. 0. 0ý 2101-2200 1. 0,
, 1. 2150.
2201-2300 0, 0. 0, 0. 2301-2400 0. 0. 0. 0. 2401-2500 0. 0. 0. 0. 2501-2600 11 0. 1. 2550. 0,000
TOTAL 2799. - 2798. ---------- ----------
1095. --
--------- 46 .2 68
TOTAL EXP. LOSS ---------------
O. F. 24
------ -------- _ 0.2 % p 0.004 TOTAL ACT. COST 141
Table (3.35)
74/4TH NUAHTEF3 DATA
C=25.00 PARAMETERS OF THE LOGNORMAL DIS. ESTIMATED BY L. S. REGRES8ION : - MEW = 4.959 SIGMA SQ. = 0.669
MEAN= 174.089 S. D. = 194.322
SSD=D. 124
AMOUNT F ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A-E)**2/E
1- 30 394. 382. 12. 186. 0,377 31- 60 452. 434. 18. 819. 0.747 61- 90 426. 406. 20. 1510. 0.985 91- 120 348. 342. 6. 633. 0.105
121- 150 272. 277. -5. -678. 0.090 151- 180 . 219. 221. -2. -331. 0.018 181- 210 154. 177. -23. -4496. 2.989 211- 240 124. 141. -17. -3833. 2.050 241- 270 105. 114. -9. -2299. 0.711 271- 300 78. 92. -14. -3997. 2.130 301- 330 75. 75. 0. 0. 0.000 331- 360 58, 61. -3. -1036. 0.148 361- 390 55. 50. 5. 1877. 0.500 391- 420 29. 42. -13. -5271. 4.024 421- 450 43. ' 35. 8. 3484. 1. f129 451- 480 24. 29. -5. -2327. 0.862 481- 510 22. 25. -3. -1486. 0,360 511- 540 24. 21. 3. 157fß. 0.429 541- 570 19. 18. 1. 555. 0,056 571- 600 14. 15. -1. -585. 0.067 601- 700 42. 37. 5. 3252. 0.676 701- 800 28. 23. 5. 3752. 1.087 801- 900 20. 15. 5. 4252. 1., 667 901-1000 17. 10. 7. 6653. 4.900
1001-1100 H. 7. 1. 1050. 0.143 1101-1200 5. 5. 0. ' 0. 0.000 1201-1300 1. 3. -2. -2501. 1301-1400 5. 2. 3. 4051. 0.200 1401-1500 0. 2. -2, -2901. 1501-1600 1. 1. 0. 0. 1601-1700 0. 1. -1. -1650. 1701-1800 0. 1. -1. -1751. 1801-1900 0. 1. -1. -1850. 1901-2000 0. 0. 0. 09 2001-2100 1. 0. 1. 205(1. . 2101-2200 0. 0. . 0. 0 . 2201-2300 (l. 0. 0. 0 2301-2400 1. (1. - 1. 2350. 2350. 1.500 ------------------------------ ---------- -- - ---- ------- TOTAL ----------
3064. ------------
3065. ---------- -----------
1059. 28.647 ---------
D--. -- -------
TOTAL EX}'. LOSS P 1U 0
---------------- 0.2 . TOTAL ACT. COST
142
C; =13.00 1a. bZi! i
t7.75`
PAF1AMr TEHS OF THE LOGNORMAL. DIS. ESTIMATED By L. S. REJRESSION MEW = 4.831 SIGMA SO.. 0.773
MEAN= 171.413 S. D. - 199.14?
SSD=0.50f
kýý --
AMOUNT E ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A_E)**2/E
1- 30 324. 298. 26. 403. 2.268 31- 60 387. 411. -24. -1092. 1.401 61- 90 345. 370. -25. -1887. 1.689
91- 120 289. 299. -10. -1055. 0.334 121- 150 253. 235. 18. 2439. 1.379 151- 180 1.87. 184. 3. 496. 0.049 181- 210 138. 144. -6. -1173. 0.250 211- 240 114. 114. 0. 0. 0.000 241- 270 93. 91. 2. 511. 0.044 271- 300 67. 73. -6. -1713. 0.493 301- 330 63. 59. 4. 1262. 0.271 331- 360 44. 49. -5. -1727. 0.510 361- 390 44. 40. 4. 1502. 0.400 391- 420 35. 33. 2. 811. 0.121 421- 450 25. 211. -3. -1306. 0.321 451- 480 26. 23. 3. 1396. 0.391 481- 510 18. 20. -2. -991. 0.200 511- 540 18. 17. 1. 525. 0.059 541- 570 22. ' 14. 8. 4444. 4.571 571- 600 17. 12. 5. 2927. 2.083 601- 700 39. 30. 9. 5854. 2.700 701- 800 19. 19. 0. 0. 0.000 801- 900 18. 12. 6. 5103. 3.000 901-1000 12. 8. 4. 3802. 2.000
1001-1100 3. 6. -3. -3151. 1.500 1101-1200 1. 4. -3. -3451. 1201-1300 1. 3. -2. -2501. 3.571 1301-1400 0. 2. -2. -2701. 1401-1500 1. 2. -1. -145(1. 1501-1600 0. 1. -1. -1550. 160 1-1700 1. 11 0. (1 O 1701-1800 0. 1. -1. -1751. 1801-1900 0. 1. -1, -1850. 1901-2000 0. 0. 0. 0. 2001-2100 1. 0.. 1. 2050. 2101-2200 1. 01 1. 2150. 2201-2300 0. 0. o. 0. 2301-2400 0. 0. 0. 0. 2401-2500 0. 0. 0. 0. 2501-2600 0. 0. 0. 0 2601-2700 0. 0. 01 01 2701-2800 0. 0. 0. 0 2801-2900 . 0. 01 p. , 0. 2901-3000 0. 0, 0. 0. 3001-3100 0. 0. 0. 0. 3101-3200 0. 0. (1. 0. 3201-3300 0. 0. (1. 0, 3301-3400 (1. 0, n, 0. 3401-350, ) 0. 0. 0, tl 3501-3600 1. 0. 1. . 3550. 1.125
, ýL ToT 2607. ----------
2604. ------
-------------
----------
-------
--ýA7ýº. __-_- -30. '133-
TOTAL EXP. LOSS
--- ------------ D. F.
P
------- u23
> 0.10 ------
T --------- = 2.2 143 TOT AL. Al T. ensT
Table (3.37)
75/2ND QUARTER DATA
C=25.00 PARAMETERS pF THE LOGNORMAL DIS. ESTIMATED BY L. S. REGRESSION ; - MEW = 4.979 SIGMA SQ. = 0.692
MEAN= 180.416 S. p. = 205.254
SSD=0.086
AMOUNT £ ACTUAL, EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS. (A-E)**2/E
1- 30 302. 309, E -7. -108. 0,159 31- 60 374. 345, 29, 1319, 2,438 61- 90 332. 323,. 9, 680, 0.251 91- 120 "277, 273. 4, 422, 0.059
121- 15Q 235. 222, 13. 1761. 0.761 151- 180 187. 179, 8, 1324. 0.358 181- 210 122. 144, -22, -4301. 3.361 211- 240 110, 116, -6, -1353. 0.310 241- 270 80*, 94, -14, -3577. 2.085 271- 300 72, 76. -4, -1142, 0.211 301- 330 47, 63, -16, -5048, 4,063 331- 360 39, 52. -13, -4491. 3,250 361- 390 40. ' 43, -3,. -1126. 0.209 391- 420 38, 36, 2, 811, 0.111
. 421- 450 29, 30. -1. -436. 0,033 451- 480 21, 25, -4. -1862,. 0.640 481- 510 30, 21, 9, 4459, 3.857 511- 540 19. 18, 1, 525, 0,056 541- 570 17. 15, 2. 1111, 0.267 571- 600 11, 13. -2. -1171. 0.308 601- 700 36. 33, 3, 1951. 0. "273 701- 800 22. 21. 1, 751. 0.048 801- 900 22, 13, 9. 7654. 6.231 901-1000 11, 91 2. 1901, 0.444
1001-1100 4, 6.. -2.. -2101. 0.667 1101-1200 3. 4, -1. -1150. 1201-1300 3, 3. 0, 01 0.143 1301-1400 6, 2, 4. 5402, 1401-1500 2, 2, 0, 0, 1501-1600 11 11 0. 0. 1601-1700 0. 11 -1, -1650. 1701-1800 1. 1, 0, 00 1801-1900 0, 1, -1. -1850. 1901-2000 1, 0, 1, 1950. 2001-2100 0, 0. 0. 0, 2101-2200 0. 0, U, 2201-2300
-------
1,
------- -----
0.
---------- ----------.. 2250,
--------- 2.000
---------- --- TOTAL
----------
2495.
------------ 2494.
---------- ----------- 2905.
---------
- 32.501
----------- D. F. =23
TOTAL EXP. LOSS --------------- 0.6 % P =0,09 TOTAL ACT. COST
144
'Table (3.3£3 }
#** 3-PARAMETER LOGNORMAL DIS. ##*
73/4TH QUARTER DATA
ESTIMATION BY MULTINOMIAL MAX. LIK ELIHOOD METHOD t- C= 14.01 MEW= 4.700 SIGMA SQ. = 0.8 03
MEAN. 150.159 S. D. = 102.166
AMOUNT P ACTUAL EXPECTED ACTUAL- EXPECTED Cl.. NO. CL. NO. EXPECTED LOSS (A-E) **2 /E
1- 30 470. 477. 1. 15. 0.002 31- 60 51R. 535. -17. -774. 0.540 61- 90 461. 443. 18. 1359. 0.731 91- 120 359. " 340. IP. 2004. 1.062
121- 150 239. 257., -18. -2439. 1.261 151- 180 21'3. 196. 17. 2813. 1.474 101- 210 148. 150. -2. -391. 0.027 211- 240 102. 117. -15. -33P2. . 1.923 241- 270 Al. 92. -11. -2810. 1.315 271- 300 5A. 73. -15. -4202. 3.0 2. 301- 330 66. 50. A. 2524. 1.103 331- 360 45. 47. -2. -691. 0.005 361- 390 39. 39. n. 0. 0.000 391- 420 35.
. 32. 3. 1216. 0.281
421- 450 . 34. 26. H. 3404. 2.462
451-- 400 20. 22. -2. -931, 0.1 R2 401- 510 29. 1R. 11. 5450. 6.722 5.11- 540 14. M. -2. -1051. 0.250 541; - 570 8" 13. -5. -2777. 1.923 571- 600 9. 11. -2. -1171. 0.364 601- 700 29.
_ 27. 2. 1301. 0.148
701- 800 1R. 17". 1. 751. 0.059 801- 900 20. 11. 9. 7654. 7.364 901-1000 .
6. 7. -1. -951. 0.143 1001-1100 4. 5. - -1. -1050. 0.200 ' 1101-1200 4. 4, ' 0. 0. 1201- 1300 1. 3. -2. -250 1. 0457.1 1301-1400 3. 2.
. 1. 1390.
1401-1500 1. 1. ' n. 0. 1501-1600 0ti 1. -1. -1550. * 1601-1? 00 1. 1. 0. 0. 1701-1800 0. . 1. -. 1. -1751. 1001-1900 0. 0. 0. 0; 1901-2000 1" 0. 1. 1950. 2001-2100 0. 0. 0. 0. 2101-2200 0. 0. 0. 0. 2201-2300 0. 0. 0. n. 2301-? _400
---
1.
-----------
0. -- - -----
1. 2350. 0.167 -------
TOTAL 3045. - -- - ' 3042.
----------- -- ------------ 5721 "
-------
- -- - `, 3-4 ý1 "t
- --------------- -- --------- -----.. ------------------- - ----
D. F. P> TOTAL EXP. LOSS P > 0.07
TOTAL ACT. COST k; CL - S, Yov D ; ̂ 7
P >0. )n
ý' 145
Table (3,39)
*** 3-PARAMETER LOGNORMAL DIS. ***
74/1ST QUARTER DATA
ESTIMATION BY MULTINOMIAL MAX. LIKELIHOOD METHOD t- C= 10.40 MEW= 4.650 SIGMA S0. = 0.1) 58
MEAN- 150. 222 S. D. = 1R7.204'
AMOUNT f, ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A-E)**2/E
"1- 30 381. 379. 2. 31. 0.011 31-"" 60. 428. 444. -16. -72R. 0.577 61- 90 351. 359. -8. -6114. 0.178 91- 120 334. 271. " 63. 6646. 14.646
121- 150 211. 203. R. 1084. 0.315
. 151- 180 133. - 154. -21. -3475. 2.864 181- 210 98. 118. -20. -3910. 3.390 211- 240 82. 91. -9. -2029. " U. "890 241- 270 54. 72. -18. -4599. 4.500 271- 300 52. . 57. -5. 71427. . 0.439 301- 330 53. 46. 7. 2208. 1.085 331- 360 36. . 37. . -1. -346. 0.027 361- 390 .
29. 30. -1. -376. 0.033 391- 420 26. 25. 1. 406. 0.040 421-. 450 22. 21. 1. 436. 0.048 451- 481) 22. 17.. 5.. 2327. 1.471 4A 1- 510 17. 15. 2. 991. 0.267 511- 540 10. 12'. -2. -1051. 0.333 541-
. 570 19. 11. R. 4444. 5.818
571- 600 4. 9. - -5. -2927. 2.778 601- ? on 26. 22. 4. 2(, n2.. - 0.727 701- 800 21. 14. 7. 5253. 3.500 801- 900 11. 9. 2.
. 1701. 0.444 901-1000 10. 6. 4. 3802. 2.667
1001-11J0 5. 4. 1. 1050. 1101-1200 2. 3. -1. -1150. 000 0 1201-1300 1. 2. -1. -1250. . 130 1-1400 2. 2. 0. 0. 1401-1500
. 0. 1. -1. -1450. 1501-1600 0. 1. -1 .- 1550.
1601-1700. 0. j. -1. -1650. X701-1800 1. 1. 0. 0 '-_------- ----------- ----------- ---------
. -------------- -
2.000 - TOTAL 2441. "2437. 4457. 49 207 .
D. F. 22 TOTAL EXP. LOSS " --------------- 1.2 a DY <o . 001 TOTAL ACT. DOST ICO:, 0.020
P <0.03
14G
Table (3040 -
### 3-PARAMETER-LOGNORMAL DIG. ***
74/2ND QUARTER DATA
EST IMATION BY MULTINOMT AL MAX. LIKEL IHOOD METHfD : - C= 14,. 94 MEW= 4.735 SI GMA SQ. = 0.760
MEAN= 151.518 S. D. = 177-. 583
AMOUNT £ ACTUAL EXPECTED ,
ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A-E)**2/E
1- 30 351. 348. -3. 46. 0.026 31- 60 380.. 411. -31. -1410. 2.338 61- 90 382. 349. 33. 2491. 3.120 91- 120 295. 272. 23. 2426. 1.945 ,
121- 150 211. 207. 4. 542. 0.077 151- 180 142. 158. -16. -2648. - 1.620 181-_'210
" 114. 122. -8, -1564. 0.525
211- 240 101. 94. 7. 1578. 0.521 241- 270 57.
. 74. -17. -4343. 3.905
271- 300 51. 59. -a.. -2.284. 1.085 301- 330 39. 47. -8. -2524. 1.362 331- 360 36. 38. -2. -691. 0.105 361- 390 25.
. 31. -2253. 1.161
391- 420 '24. 25. -1. . -406. . 0.040 421- 450 27. 21. 6. 2013. 11714 451- 480 18. 17. 1. ' 466. 0.059 481- 510 21., 15. 6. 2973. 2.400 511- 540 17. 12. 5. 2627. 2.083 541- 570 12. 10. 2. 1111. 0.400 571- 600 11. 9.. 2. 1171. 0.444 601- 700 30. 21.. 9. 5854. ` 3.857 701- 800 . 13. . 13. 0. , 0. 0.000 801- 900 11. R. 3. 2551. 1.125 901-1000 4. 6.
-2. -1901. 0.667 1001-1100 7. 4. 3. - 3151. 1101-1200 0. 3. -3. -3451. " 1201-1300 ,
1. 2. -1. -1250. x"000
1301-1400 1. 1. 0. 0. 1401-1500 0. 1. -1. -1450. 1501-1600 " 1. 1. 0. 0.
, 1601-1700 0. 1. -1. -1650. 1701-1800 0. 0. 0. . 0.
-180 1-1900 " 1. 0. 1. _
1850. 0.667
---------- TOTAL
----------- 2383.
--------- 2380.
------------- ----------- 3626.
-------- 31.248
----------- ----------- --------- -----; D. F. 24
TOTAL EXP. LOSS . _-_-_ ---------- = 1.0 % ..
P 0.07 TOTAL ACT. CAST 1; OL - SP,, lI
. NOV D 0'. 01 i P>0�20
147
Table (3.4,1)
xý` 3-PAHAMETEH LOGNOF8 AL DIS.
74/3H0 ((UARTER DATA
ESTIMATION BY MULTINQMIAL MAX. LIKELIHOOD METHOD ; -- Cs 15.67 MEW- 4. ß21 SIGMA' 5Q. = 0.759
MEF, rd= 165.694 S. D. = 193.304
-AMOUNT £ ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO, EXPECTED LOSS (A-Ej**2/E
1- 30 362. 359. 3. 46. 0.025 31- 60 427. 446. -19. -865. 0.809
61- 90 383. 395. -12, -906. 0.365 91- 120 356, 318. 38. 4009. 4.541
121- 150 283. 249. '34. 4607. 4.643 151- 180 194. 194. 0. 0. 0.000
181- 210 137. 152. -15. -2933. 1.480 211- 240 97. 120 -23. -5186. 4.408
241- 270 86. 95. -9: -2299-. 0.853 271- '3.00 71. 76. -5. -1427. 0.329 301- 330 64.
. 62. 2. 631. 0.065
331- 360 45. 50. -5. -1727. 0.500
361- 390 44. 41. 3. 1126. " 0.220 391- 420 25. 34. -9. -3649. 2,382 421- 450 20'. 28. -2. -871. 0.143 451-. 480 22. 24. -2. --931. 0.167 481- '510 25.. 20.. 5. 2477. 1.250 511- 540 14. 17. -3. -1576. 0.529 541- 570 14. 15. -1. -555. 0.067 571- 600 17. 12.. 5. 2927.. 2.083 601- 700 32. 30. 2. 1301. ' 0'. 133
701- Boo 34. 19. 15. 11257. 1.1.042 801- 900 17. 12. 5. ' 4252. - 2.083 901-1000 4. 8. -4. ' -3802. 2.000
. 1001-1100 9. 6. 3, 3151. 1.500 1101-1200 4.0 4. , 0. 0. 120.1-1300 0. 3. -3. -3751. 1.286 13011-1400 1. 2., -1. -1350. 1401-1500 3. 7, ". 1. . 1450. 1501-1600 0. 1. -1. -1550. 1601-1700 1.
. 1.. 0. 0.
1701-1800 0. 1. -1. -1751. 1801-1900 0. 1. -1. -1850. 1901-2000 0. 0. 0. 0. 2001-2100 0. 0. 0. . 0. 2101-2200, '1. U. 11 2150. 2201-2300 0. 00 0. 0.
.. 2301-2400 0. 0. 0. 0. 2401-2500 ' 0. 0. 0. 0.
2501-2600 ----
1. -----------
0. ----------
1. ------------
2550. -----------
0.125 -------- ------
TOTAL 2799. 2797. 4956. 43.828
D . F. ?3 TOTAL EXP. LOSS
---... -- ---------ý 1 .1 aro P 0.006
TOTAL ACT. COST Y0i - SMIRNOV D 0.01
P>0.05
148
as ble 3.42)
3-PARAMETER LOGNOft
74/4TH QUARTER
ESTIMATION BY MULTINOMIAL MAX. C= 18.18 MEW= 4.878
MEAN= 174.458
L1AL DIS. * -*
DATA
LIKELIHOOD METHOD : - SIGMIA SQ. = 0.766
S. D. = 206.685
AMOUNT £ ACTUAL EXPECTED ACTUAL- EXPECTED' CL. NO. EXPECTED LOSS (A-E) **2 /E
1- 30 394. 393. 1. 15. 0.003. 31-, -60 452. 462. - -10. -455. 0.216. 61- 90 426. 414. 12. 906.. 0.34E 91- 120 348. 338. 10. 1055. 0.296
121--150 272. 269. 3. 4'06. 0.033 151- 180 219. 212. 7. 1158. 0.231 181- 210 154. 168. -14. -2737. 1.167 211- 240 124. 134. -10. -2255. 0.746 241- 270 105. 108. -3. -767' 0.083
, 271- 300 78. 87. -9. -2569. 0.931 301- 330 75. 71. 4. 1262. 0.225 331- 360 58. 59. -1. -346. 0.017 361- 390 55. 49. 6. 2253. 0.735 391- 420 29. 40.
. -Il. -4460. 3.025 421- 450 43. 34. 9. 3919. 2.382 451- 480 24. 29. -5. -2327. 0.862 481- 5.10 22. 24. -2. -991. 0.167 511- 540 24'. 21. 3. 1576. 0.429 541- 570 19.. 18. 1. 555. 0.056 571- 600 14. 15. -1. -585. 0.067 601- 700 42. 37. 5. 3252. 0.676 701- 800 28. 24. 4. 3002, 0.667 801- 900 20. 16. 4.
. 3402. 1.000
901-1000 17. 11. 6. 5703. 3.273 1001-1100 8. 7. 1. 1050. 0.143 1101-1200 5. 5. 0. . 0. 0.000 1201-1300 1, 4. -3. -3751. 1301-1400 5.
, 3.
, 2. 2701. 0.143
1401-1500 0. 2. , -2. -2901. 1501-1600 1. 2. ' =1. -1550. 1601-1700 0. 11 -1. -1650. , 1701-1800
_ 0.
. 1. -1. -1751. 1801-1900 0. 1. -1. -1850. 1901-2000 0. 1. -1. -1950. 2001-2100 1. 00 1. 2050. 2101-2200 0. 0. ` 0. 0. 2201-2300 ' 0.. 0. 0. 0. 2301-2400
11 -----------
1. -----------
0.
----------- 1.
------------ 2350. ",
------- 3,1'25
TOTAL ----------
3064. -----------
3060. ----------- ------------
-- 3722.
-- "
------------ 21.044
------- DF. . ' = 2/1 TOTAL. EXP. LOSS - P > 0.10
----- ----------- n ^7 ct i ----- -- u ", / rr
TOTAL ACT. CnST SI: MNOv D =0-C, (17
P >0.20
75/ *, S i QUAK i CR DATA Hin {7 A7, `
ESTIMATION i'v' MULTINOMRIAL MAX. LIKELIH0ü0 METHOD C= 21.00, N", EW 4.915 SIGMA SQ.. 0. -4n 1 ̂ ! MEAN= 173.207 S. D. = 197.536
AMOUNT C ACTUAL EXPECTED ACTUAL- EXPECTED CL. N0. CL. NO, EXPECTED LOSS (A_E)**"2/E
1- 30 324. 325. -1. -15. 0.003 "31- 60 387. 383. 4. 182. 0.042 61- . 90 345. 352. -7. ' -52R. 0.139 91- 120 289. 292., -3. -316. 0.031
121- 150 "253. 234. 19. 2574. 1.543 151- 180 187. 186. 1. 165. 0.005 181- 210 138. 147. -9. -1759. 0.551 211- 240 114. 117, -3. -677. 0.077 241- 270 93. 94. - -1. - -256. 0.011 271- 300 67. 76. -9.. -2569. 1.066 301- 330 63. 62. 1. 316. 0.016 331- 360 44. 51. -7. -2418. 0.961 361- 390 44. 42. 2. " 751. 0.095 391- 420 35. 35. 0. 0.. 0.000 421- 450 25. 29. -4. -1742. 0.552 45"1- 480 26. 24. 2. * 931. 0.167 481- 510 18. 21. -3. -1406. 0.429 511- 540 18. 17. 1.. 525. 0.059 541- 570 22. 1"5. 7. 31388. 3.267 571-. 600 17. 13. 4. 2342. 1.261 601- 700 39. 31. 8. 5204. 2.065 701- 800 19. 19. 0. 0. 0.000 801- 900 18, 13. 5. 4252. 1.923 901-1000 12. ß. 4. 3802. . 2.000
1001-1100 3. 6. -3. -3151. 1.500 1101-1200 1. 4. -3. -3451. 1201-1300 1. 3. -2. -2501. 3.571 1301-1400 0. 2. -2. -2701. 1401-1500 1. 2. -1. -1450. 1501-1600 0. 1. -1. -1550. 1601-1700 1. 1. 0. 0. 1701-1800 0. . 1. -1. -1751. ,. 1801-1900 0. 1. -1.. -1850. 1901-2000 0. 0. 2001-2100 1. 0. 1. 2050. 2101-2200 1.. 0. 1. 2150. 2201-2300 0. 0. 00. 0. 2301-2400 0. 0,, 01 0. 2401-2500 0. 0.. 0. 2501-2600 0. 0. 2601-2700 0.
. 0. . 0. 0ý 2701-2800, 0.
. 0. 0. 0.
2801-2900 . 0. 0. 0. 0. 2901-3000 0. 0. 0.
. 0. 3001-3100 0. 0. 01 3101 -3200 0. 0. 0. 0 3201-3300 0. 0. 0. .
0. 3301-3400 0. . 0. 0. 10», , 3401-3500 0. 0. 0. 3501-3600 1. 0. 1 3550 ..
---- ----------- ----------- .
-- TOTAL
__-.... ------ 2607.
----------- 2{07.
-----------
--
-----------
----------- 2510.
-----
----- -- -
TOTA! EXP 1 05S X4... ------
1?; C7 . (ß. v07 D .F. 23 ` p>G. '. 20 "----- ---------= 0 .6 % p C. 1ý TOTAL P. C [, r'iC, T
. 44) Table 0
---------- ---- TOTAL ACT. COSTr i: OL - ;; MU! "lo
151
" #'"` 3-PARAMETER LOGNORMAL DIS. ***
75/2N0 UUAP+ T ER DAT A
ESTIMATION BY MULTINOMIAL MAX,. LIK ELIHOOD METHOD ; - C- 15.16 MEW= 4.872 SIGMA SQ. = 0.8 07
MEAN- 180.409 S. D. = 217.852
AMOUNT £ ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A-E)**"2/E
1- 30 ; 302. 302. 0. 0. 0.000 31-. 60 374. 376. -2. -91. 0.011 61- 90 332-. 337.
.. -5. -378. 0.074 91- 120 277. 275. 2, 211. 0.015
121- 150 235. 218. 17. 2304. 1.326 151- 180 1137.
. 173. 14. 2317, 1.133
"181- 210 *
122. - 137. -15. -2933. - 1.642 240 2-11 110, 110. 0. o. 0.000
241- 270 f; 0, 89, -'-9. -2299. 0.910 271- 300 72. 72. 0.
, 0. 0.000 301- 330 47. 59. -12. --3786. 2.441
-331--, 360 39. 49. -10. -3455. 2.041 " 361- 390 40. 41. -1. . --376. 0.024
391- 420 38. 34. 4. . 1622. 0.471 421- 450 19. . 29. 0. 0. 01000 451- 480 219 '24" -3. " -1396. 0.375
. 481- 510 30. . 21* 9. 4459. 3.857 511- 540 19. 18. 1. 525. "0.056 541- 570 17. 15. 2. 1111. 0.267 571- 600 11. 13. -2. -11710 0.308 601- 700 36. 339 3. 1951. 0.273 701- 800 22. 21. "1. 751. 0.048 801- 900 22, 14. .
H. 6804. 4.571 901-1000 11. 10. 1. 951. 0.100
1001-1100. 4. 7. -3. -3151. 1.286 1101-1200 3. 5.
- -2. -2301. O. 11C
1201-1300 3. 4. -1. -1250. 1301-1400 6. 3. 3. 4051. 0.71 1401-1500 2. 2. 0, !1 1501-1600- 1. . - 2. -1. -1550. 1601-1700 0. 1. -1. -1650. 1701-1800 1. 1. 0. 0"40 "" 1801-1900 - (3. 1.. -1. -1850. 1901-2000 1. 1. - 0,
, 06
2001-2100 " 0. 0. 0� 0l 2101-2200 0. " 0. .O0 U 2201-2300 1. 0. 1. ." 2250.
0.1700 ------------ 0------------------- -------------------------
-------
TOTAL
-----------
2495.
---------- 24970
---------- ----------- 1669,
--------------- 23.897 --------
TOTAL ERP. LOSS 0, F.
p 24
.ý0.10
.ý
*** S-PARAMETER LOGNORMAL DIS. *x
PREDICTION OF 24/KITH QUARTER CLR1? MS COST USING 73/4TH QUARTER PArIAM TERS
C= 14. P MEW'= 4.700 SICMA2=C. Rh3
ti INFLATION HATE I=1R. 2°+ CALCULATED PROM GENERAL INDEX OF PETAIL PRICES
-PREDICTION PARAMETERS ARE .- C= 16.5 MEW- 4.867 SIGMA2=0. H03 MEAN CLAIM AMOUNT=e 177.62 S. D. =215.53
ACTUAL 74/4TH PARAMETERS :- C= I R. 2 MEW- 4.87R SIGMA2=0.7F, 6 MEAN CLAIM AMOUNT= 174.47 S. D. =2n6.72
AMOUNT £ ACT. NO. EXP. NO. A-E EXP. LOSS (A7E)*"*2/E
. 1- 30 394. 1 394.. 1 0. 0. 0.000 31- 61 452. 464. -12. -546. 0.310 fý1- 0 426. 412. 14. 1057. 0.476 91- 120 348. V335. 13. 1371. 0.504
121- 150 272. 266. 6. 813. 0.135 151- 180 219. 210. 9. 14 P9. 0.386 181- 210 154. 167. -13. -2541. 1.612 211- 240 124. 133. -9. -2029. 0.609 241- 270 105. 107. -2. -511. 01-037- 271- 300 ? S. 87. -9. -2569. 0.931 301- 330 75. 71. 4. 1262. 0,225 331- 360 `F8. 59. -1. -346. 0.017 361- 390 55. 49. 6. 2253. 0.735 391- 420 29. Al. -12. -4866. 3.512 421- 450 43. 34. 9. 3919. 2.382 451- 480 24. 29. -5. -2327. 0. R62 V 481- 510 22. 25. -3. -1486. 0.360 511- 540 24. 21. 3. 1576. 0.429
. 541- 570 19. 18.
. 1'" 555. V 0.056
571- 600 14. 16. -2. -1171. 0.250 601- 700 42. 39. V 3. 1951. 0.231 701- 800 2R. 25. 3. 2251. 0.360 801- 900 20. 17. 3. 2551. 0.529 901-1000 117. 12. 5.
. . 4752. 2.083
0 1001-110 A. R. 0. f1. 0.000 , 1101-1200 5. 6. -1. ' -1150. 0.167
1201-1300 1, 4. -3. -3 151 . 1301-1400 5. 3. 2. 2701. 1401-1500 0. 2. -2. -2901. 1501-1600 1.. 2. -1. -1550. 1601-1700 0. 1. -1. -1650. 1701-1800 0. 1. -1. ' -1751. 1801- 1900
, 0. 1. -10 -1 P50.
1901-2000 0. 1. -1. -1950. 2001-2100 1. 1. 0. 0. 2101-2200 0. U. C;. 0. 2201-2300 0. 0. 0. U. 2301-2400 V 1. 0. 1.. 2350.
------. ----- TOTAL
------------
-------- 3064
--------
---------- 3061
----------
-------
-------
---------- -4093.
----------
---------- 16.599
----------
CHI SO. STA, T ."= 20,67 > 0-F--28
TOTAL ACTUAL COST = 53? 707. TOTAL EXPECTED COST =
TOTAL EXP. LOSS
TOTAL ACT. COST 152
P=C;. FýS
t Ti 0.0 i' 7 ß'. 2O
sable (3,46) PREDICTION OF 75, % 1ST (, MARTER CLAIMS COST USING ?L /1"- QL! RTER M JLT. PIiAXKIK. PARAMETERS ;
C= 10.4 MEVI= 4,650 SIGMA2=0.050 INFLATION PATE I_20.3°- CALCULATED C. pq
'GENERAL IN:? EX OF RETAIL PRICES
i
hhl-
PREDICTION PARAMETERS ARE "- L'- 12.5 MEW= 1,835 SIGMA2=0.85P " MEAN CLAIM AMUUNT= 18U. 71 S. D. =225.20
ACTUAL 75/1ST PARAMETERS C= 21.1 MEW= 4.9.15 SIGMA2=0. -710 MEAN CLAIM AMOUNT= 173.32 3. D. a197.69
AMOUNT 'f ACT. 140. EXP. NO. A-E EXP. LOSS (A-E)**2/E" 1- 30 324. 321. 3.. 46. 0.028
"31- 60 387. 404. -17. -774. 0.715 61- 90 345. 355. -10. -755. 0.282 . 91- 120 2A9. 285. 4. 422. 0.056
121- 150 253. 225. 28. 3794. 3.484 151- 190 1817. 177. 10. 1655. 0.565 1R1-'210 138. 140. -2. -391. 0.029 211- 240 114. 112. 2. 45-1. 0.036 241- 270 93. 90. 3. 767. . 0.100 271- 300 67. 73. -6. -1713. 0.493 301- 330 63. 60. 3. 947. 0.150' 331- 360 44. 50. - -6. -2073. 0.720 361- 390 44. 42. 2. 751. 0.095 391- 421) 35. " 35. 0. 0. 0.000 421- 450 25. 29. -4. -1742. 0.552 451- 4R0 26. ' 25. 1. 466. 0.040 481- 510 in. 21. -3. -1486. 0.429 511- 541) 18. 18. U. " 0. 0.000* - 541- 570 22. 16. 6. 3333. 2.250 571- 600 17. 14. 3. 1756. 0.643 601- 700 39. 34. ' 5. 3252. 0.735 701- 800 19. 22. -3. -2251. 0.409 801- 900 18. 15. 3. 2551. 0.600 901-1000 12. 10. 2. 1901. 0.400
1001-1100 3. 7. -4. -4202. 2.286 1101-1200 1. 5. -4. -4602. 3.200 1201-1300 1. 4. -3. -3751. 1301-1400 0. 3. -3. -4051. 1401-1500 1. 2. -1. -1450.
"1501-1600 0. 2. -2. -3101. 1601-1700 1. 1. 0. 0. 1701-11300 0. 1. -1. -1751. 1801-1900 0. 1. -1. -1850. 1901-2000 0. 1. -1. -1950. 2001-2100 1. 1. 0. 0. 2101-2200 1. 0. 1. 2150. 2201-2300 0. 0. 0.1 0. 2301-2400 0. 0. 0. 0. ` 2401-2500 0. 0. 0. 0. 2501-2600 0. 0. . 0. 0. 2601-2700 0. 0. 0. 0, 2701-2900 0. 0. 0. 0. 2801-2900 0. 0. 0. 0. 2901-30001 0. 01 01 0. 3001-3100 t3. 0. 0. 0. 3101-3200 U. 0.. 0. 0. 3201-3300 0. 0. 0. 0. 3301-3400 0. 0. 0. 0. 3401-3500 0. 0. 0. 0. 3501-3600
-
1. 0. 1. 3550. ---- -------
T0TAt. --------
26;; -7
-----------
2601
----- -----------
-I tý 1 . ", 2. ----------
1 f1.29'7
Table (3.46) - continued
CHI SO. STAT. = 25.76 , D. F. = 2E)
TOTAL ACTUAL COST = 452059-- TOTAL EXPECTED COST - 462161.
TOTAL EXP. LOSS
--------------- - -2.23 °c TOTAL ACT. COST
R
y
P>0.50
KOL - S"IIR? d0V I= 0.01
- P> 0.20
Ta! C ## j-PARAMEiE. I IA,
_ : 1ý::, **s.
1i
hkhl-
'Pi1E DICTION OF ? 5/2ND QUARTE9 CLAIMS COST. USING 74/2ND WUARTER MULT. MAXLIK. PARAMETERS
Ca 14.9 MEW= 4.735 RIGMA2so. 76n INFLATION RATE I=24.3% CALCULATED FROM : GENERAL INDEX dF RETAIL PRICES
PREDICTION PARAMETERS ARE :- C= 1R. 5 MEW- 4.953 SIGMA2 =. 760 MEAN CLAIM AMOUNT= 108.44 S. D. =220. x1"
ACTUAL 75/2 ND PARAMET ERS C= 15. 2 MEW= 4.072 SIGMA2=0. At)7 MEAN CLAIM AMOUNT= 180.29 S. 0. =217.79
AMOUNT f ACT. NO. EXP. NO. A-E EXP. LOSS (A-E)**2/E 1-
, 30 302. . 279. 23. 356. 1.896
31- 60 3? 4. 349. 25. 1137. 1.791 , (, )1- 90 332. 325. 7. 528. 0.151
91--120 277. 273. 4. 422. 0.059 121- 150 235. 222. 13. 1761. 0.761 151- 180 187. 170. 9. 1409. ' ' 0.45 5 181- 210 122. 143. -21. -4105. 3.0P4 211- 240 110. 116. -6. -1353. 0.310 241- 270 00.. 94. -14. -3577. 2.085 271- 300 72. 77. -5. -1427. 0.325 301- 330 47. 63. -16. -5040. 4.063 331- 36n 39. 52. -13. -4491. 3.250 361- 390 40. 44. - -4. -1502. 0.364 391- 420 30. 37. 1. 406. 0.027 421- 450
_29. 31. -2. -871. 0.129
451- 400 21. 26. -5. -2327. 0.962 481- 51Q 30. 22. A. 3964. 2.909 511- 540 19. 1q. 0. 0. , 0.000 541- 570 17. 16. 1.. 5550' . 0.063 571- 600 11. 14. -3. -1755. 0.643 601- 700 36. 35. 1. 651. 0.029 701- 900 22. 23. -10 -751. 0.043 001- 900 22. 15. 7. 5953. 3.267 90 1-1000 11. 10. 1. 951. 0.1(10 001-1100 4. 7. -3. -3151. 1.206
1101-1200 3. 5. -2. -2301. 0.300 1201-1300 3. 4. -1. -1250. 1301-1400 6. 3. 3. 405 1. 1401-1500 2.. 2. 0.. 0. 1501-1600 1. 2. -1. -1550. 1601-1700 0. 1. -. 1. -1650. 1701-1000 1. 1. 0, 0, 1001--1900 0. 1, 1901-2000 1. 1. 0. 2001-2100 . 0.. 0. 0. 0. 2101-2200 0. 0. n. 0. ," 2201-23(10 11 . 0. 1,
. 2250,
TOTAL . ------------
2495
----------
2490
------- ------- -14407.
.
----------
. _--20.05(_-
-------------
GHI ", Q. STAT. _ 29a9 , D. F. = ý",
T(. UT'A'L AGTUAL GQ; 3T TOTAL EXPECTED COST - 46379.
TO1AL EX! ". LD,: S
-3 TOTAL Ai; T_ý .? _?
"ý' 155 C. ST
p>o, o OL -P C' " C'
T_n. ý
CHAPTER 4
The Weibull Distribution
4.1 Introduction
The Weibull distribution is named after Waloddi Weibull, the
Swedish physicist, who in 1939 derived it empirically from practical
considerations of the size effect on failures in solids. He used this
distribution as a model for the breaking strength of materials.
The Weibull is a flexible distribution which is related to
the exponential family. For a suitable choice of parameters, it is
positively skewed with a tail longer than that of the exponential
distribution. Over the years, it has been used to represent the
distribution of random outcomes of various phenomena in science and
engineering. Weibull (1951) has fitted this distribution to data on
1- yield strength of Bofors steel ;
2- size distribution of fly ash ;
3- fibre strength of Indian cotton ;
4- statures for adult males, born in the British Isles ;
etc.
In recent years the Weibull distribution has become one of the most
widely used statistical distributions in the fields of reliability, life
and fatigue testing of engineering systems and their components. A
bibliography on this distribution and its applications appears in
Johnson and Kotz (1970).
156
The (negative) exponential distribution has been used in
general insurance, and in particular in fire insurance, as a model for
the distribution of claim armunts (see, for instance, Ramachandran (1970)).
It is important to see how the more flexible Weibull distribution
would perform in this respect. In addition, the fact that the Weibull
has successfully represented the distribution of positively skewed
random variables, in various fields, encourages us to consider its
application as the model for the distribution of claim amounts in
general insurance. Therefore, in this chapter, the Weibull distribution
will be initially defined and some of its properties will be mentioned.
Then a graphical procedure will be suggested for testing whether a
particular set of data may be regarded as a sample from a
Weibull population. The accidental damage data will then be tested
accordingly. In section 4.6 the least squares and multinomial maximum
likelihood methods of parameter estimation from grouped data will be
considered. These methods will then be used to fit the Weibull model
to the samples of accidental damage data. The effects of the inflation
on the parameters of the model will be studied in section 4.8. Finally,
the conclusions of this chapter will be discussed in section 4.9.
4.2' Definition
A random variable X is said to have a 3-parameter Weibull
distribution if its distribution function, denoted by
W(x ; C, A, B) is of the form .
157
P(X x) =1- exp [_ (x - ClB
;x>C (4.2-1) J
=0
for A>O, B>0, -oo <C<°
This distribution is related to the standard exponential distribution
via the transformation
Y= (X - C\B (4.2-2)
which leads to P(Y <, y) =1- e-y ,y>0
Hence Y has the standard exponential distribution (i. e. with mean = 1).
The. probability density function (p. d. f. ) of X, denoted by fw(x; C, A, B),
is
= B/x-CB-1 x-CB fW(x, C, A, B) A`A) exp -`Aj; x> C (4.2-3)
It is apparent that C is a location parameter below which the values of
X are not realized. C is often called the threshold parameter. A
and B are scale and shape parameters respectively. If in (4.2-1)
and (4.2-3) we put C=0 we will have the 2-parameter Weibull
distribution and probability density functions respectively.
4.3 Properties of the Weibull Distribution
In this section we will briefly report some properties of the 3-
parameter Weibull distribution. More details on this can be found in
Johnson and Kotz (1970), Bury (1975) or Mann et al. (1974). The
properties of the 2-parameter distribution will be obtained by putting C=0 in the following.
158
The Ii`eibull distribution can represent various shapes through
the parameter B. In particular, when B<1 the p. d. f. is monotone
decreasing, with a tail longer than that of the exponential distribution,
and there is no mode. For B=1 this distribution becomes equivalent
to an exponential distribution with C and A as location and scale
parameters respectively. For B>1 the p. d. f. of the Weibull
distribution is unimodal, with the node at
xC+A[ (B - 1)/B] 1/B
defined in (4.2-2), has the standard exponential distribution.
In this case the p. d. f. is positively skewed for B<3.6
(approx. ), and negatively skewed for B>3.6 (approx. ). For B=3.6
(approximately) the shape of the Weibull distribution will be similar
to that of a normal distribution.
To find the moments of X about zero we use the fact that Y, as
" Let Z=XAc therefore Y= ZB and X=C. + AZ.
The rth moment of Z about zero is
9 al
R A:
E (Zr) =E [(ZB)r/B
=E (Yr/B) =B th moment of Y about zero
hence from the theory of exponential distribution it follows that
(4.3-1)
E (Zr) =r(+ 1) 11 (4.3-2)
Therefore moments of Z and hence of X can be obtained. For r=1
we have E (Z) =r(,, + 1)
Therefore the mean of X. will be
E (X) =C+A r( + i) (4.3-3)
159
and by using r=2 in (4.3-2) we can derive the variance of X as
' var(X) -A {(2
+ 1) - [r (B
+ 1) ]2} (4.3-4)
The coefficients of skewness and kurtösis can be obtained as
complicated expressions involving the Gamma functions. These are
derived in Bury (1975).
The median of the distribution can easily be shown to be at
x=C+A (log 2)1/B (4.3-5)
The quantile of order q is at
xC +A log 1 1/B (4.3-6)
i-a
In many applications there is usually no explicit theoretical
justification for the use of the Weibull distribution as a model.
However, if a negative exponential model can be justified, or
shown to be reasonable, for a random phenomenon, then its replacement
by a Weibull distribution will. allow greater flexibility in the model.
It is with these remarks in mind, and the fact that the exponential
distribution has been used as a model for claim amounts, that we
study the Weibull distribution in this chapter.
4.4 The Graphical Test for the Weibull Distribution
We explain this test for the 3-parameter case. For the 2-
parameter Weibuil distribution the argument holds by putting C=0.
160
Let uS denote the cumulative distribution function
W(x; C, A, B) simply by W(x) where
11 (x) =1- exp -ACB]
Hence
1- W(x) = exp (xAC)B
A
by taking natural logarithm twice on both sides and simple manipulations
we can show that
log log 1 -1 «(x =B log (x - C) -B log A (4.4-1)
Therefore the locus of the points (
log(x - C), log log 1 -1W W(X7) is
a straight line. This provides us with a means of testing if a sample
of data is from a Weibull population. Let us assume that we are given
a sample of n independent observations on the random variable X. At
a particular value :: i we define the sample empirical distribution
" function F(xi), which is an estimate of ta(xi), as
F(xi) = the proportion of observations 4 xi (4.4-2)
Therefore if the value of C is knowm, for the 3-parameter case, and
the sample is from a Weibull population we would expect the points
(i. og(xj - C) , log log
1-l, Xi )
to lie approximately on a straight
line - say, line 1 in figure (4.1). The parameters A and B can be
estimated from the slope and intercept of this line. The use of a
special graph paper called the N'eibull probability paper makes the tasks
of calculations, for plotting the points, and of estimation of the
parameters easier. On this graph paper, one of the axes is scaled
161
r4
bO 0 ho 0
r-i
=C
>C
<C
log (X - C)
Figure (4.1) - Graph of the points (log(x
- C), log log 1- F(x1)
logarithmically while the other is graduated such that the proportions
F(x) are converted to log log 1 _1F x. Therefore, for plotting,
only the points (x - C, F(x)) need to be calculated. There are
special scales provided on the paper which allow, after a successful
fit of a straight line to the points by eye, the estimates of the
scale and shape parameters to be read from the graph.
So far we have assumed that the value of C is known. If this is
not so then the values of x-C cannot be calculated. In this case C
has to be estimated beforehand. If we use Cl for C where Cl >C
(i. e. if we overestimate C) then, because this will have a larger
impact on log(x - C) for smaller values of x, we will find that the
points lie on a convex curve - plot 2 in figure (4.1). If we
underestimate C (i. e. use C' < C) then a concave curve will be obtained
plot 3 in figure (4.1). We faced the same problem in section 3.11
on testing for 3-parameter log normality The procedure which was
13
162
applied in that section can be adopted here. Therefore we try
various values of C and for each value plot the points
1 (log (x - C) , log log i_ `F;
X,. By observing the resulting
plots we decide on the best value of C which rectifies the points to
lie approximately on a straight line. If, in fact, such aC exists
then we can conclude that our sample is from a 3-parameter Weibull
population. If necessary, the parameters A and B can then be estimated,
as before, from the slope and intercept of the line.
The initial value of C to be tried always is C=0 which will
either indicate that the sample is from a 2-parameter Weibull population
or, alternatively, that by choosing C=0 we are underestimating or
overestimating C. If it indicates underestimation then the mdel will
involve a positive location parameter and its distribution function
can be expressed as
W(x; C, A, B) =1- exp L(XA C) B
(4.4-3)
If C=0 indicates overestimation then the model may involve a negative
location parameter. Hence, for the sake of avoiding the use of
negative values for parameter C, the model can be expressed as
W(x; -C, A, B) =1- exp [-( X+ C} BI
;C>0 (4.4-4) AJ
4.5 The Weibull Graphical 'Test ' on 'AD 'Data
We wrote the computer program P13 to use a sample of grouped
data, as input, and to plot the points (log(x
)-Cs log log g 1- x)
163
for various values of C. For each sarple of accidental dai-age data,
as presented in tables (1.1) to (1.7), values of C=0,10,15,20
and 25 were supplied to the program. For each C, the program then
calculated and plotted the resulting array of points. The graphs
for each of the seven samples of data are presented in figures
(4.2-a) and (4.2-b). It can be seen that for each sample, C=0
has produced a concave curve with a marked deviation from a straight
line in the lower values of x. Therefore C=0 has underestimated
the true value of C. On the other hand, for C= 25, for each sample,
a convex curve has been produced indicating an overestimation of C.
For each sample C= 10 also indicates an underestimation of C.
Therefore the true value of C, in each case, must lie between 10 to 25.
In fact it is apparent from the graphs that C= 15 or 20 are about the
best choice to make the array of points lie on a straight line.
There are slight deviations from the straight line in the upper tail
values for each sample. These are, as in the case of the lognormal
distribution, due to lack of sufficient observations in the long tail
of the sample histogram.
We feel justified that a 3-parameter Weibull model with a positive
location parameter C, in the range of 10 to 25, and cumulative
distribution function of the form (4.4-3) may be used to represent the
distribution of our accidental damage data.
We Imow that some policy holders do not claim for small amounts
because their no claim bonus is worth more than the amount they would
recover by a claim. Considering that our proposed Weibull model involves
terms of the form x-C we may interpret parameter C as the anount below which claims are not made.
164
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4.6 Estimation of the Parameters of the leibull Distribution
The estimation problem for the Weibull distribution has been
dealt with by various authors. However, rast of the concentration
appears to have been on estimation from ungrouped data and for the 2-
parameter distribution. Kao (1956) considered the methods of least
squares and maximum likelihood for tmgrouped data when the scale
and shape parameters, A and B respectively, are unknown. In
addition, in Kao (1958) he dealt with estimation from grouped data, 2-
parameter case, by the methods of nnultinomial maximum likelihood
and minimum Chi-square.
The method of moments can be used which involves equating the firs:.
two (or three in the 3-parameter case) s Vle moments to their
corresponding population values and solving the resulting equations
for the unknown parameters. Dubey (1963) reports that he has found
" the asymptotic efficiencies for the roment estimators of the location
and scale parameters when the shape parameter is known. These, he
states, depend on the value of the shape parameter and are quite high
in some cases.
The method of quantiles may also be used for estimation. This
consists of equating two (or three in the 3-parameter case) sample
quantiles to their corresponding population values. Dubey (1967)
has applied this method for estimating parameters of the 2-parameter
distribution. He states that the 17 and 97% quantiles give an
estimate of the shape parameter which is asymptotically 66% efficient
when compared with the maximum likelihood estimator. The 40 and 82%
quantiles estimate the scale parameter with an asynptotic efficiency
of 82%.
165
A relatively quick and easy method of estimating the parameters
of the Weibull distribution is the graphical method which we explained
as part of the Weibull graphical test in section 4.4. This method
has been fully explored by Kao and we can, for example, rifer to
Kao (1959).
Haan and Beer (1967) deal with estimation from ungrouped data
by the ordinary maximum likelihood method for the 3-parameter Weibull.
In this case a system of equations involving powers of the unknown
parameters need to be solved iteratively.
When data only in grouped form is available the estimation problem
usually becomes more complicated. It is possible to assume that the
observations in every interval are concentrated at the mid-point of
that interval and hence use the estimation procedures for ungrouped
data. This assumption, cven when reasonably justified, results in
some loss of efficiency and may introduce certain biases in the
estimators. Therefore, as we showed in the 2-parameter lognormal
case, it is best to use methods which are directly suitable for
grouped data. Of these, the most efficient method is the multinomial
maximum likelihood. This, as we mentioned earlier, was considered
for the 2-parameter Weibull by Kao (1958). In the present work we
will extend this method to the case when all three parameters are
unknown. In addition, we will consider the method of least squares
and will propose a computing technique for estimating all three unknown
parameters.
Let us assume that we have a sample of grouped data where n inde-
pendent random observations on a random variable X (in cur case, the
claim amount) have been grouped according to their size into k mutually
exclusive intervals. Further, let ni be the number of obsertiaticns
166
(claims) in the class interval (xi_1 , xi) for i=1,2..., k such
that k
n= nl i=1
Also, let us assume that X has a 3-parameter Weibull distribution
with the cumulative distribution function of the form
W(x; C, A, B) =1- e{- (x
A C)B
x>C (4.6-1)
We define the sample empirical distribution function as
F(x) = the proportion of observations <x (4.6-2)
In. the following sections we shall use the above notation. Although
we will be considering the estimation problem of the 3-parameter
ti; Weibull distribution, the methods can be directly used for the 2-parameter
Weibull by putting C=0.
4.6.1 The Method of Least Squares
We showed in section 4.4 that if our sample is from a Weibull
population then the points (log (x1 - C), log log 1 -1F xi
)'
for i=1, ..., k-1 are expected to lie on the straight line
log log =B log(x - C) -B log A (4.6-3)
It should be noted that we cannot use xk because F(xk) =-1 and hence
will be ind3terninate.. To find the unknown parameters we 1-F (xk)
can use the least squares regression technique which consists of
167
(1
riinindsing SSD, the stur. of squares of deviations, where
k-1 SSD =
[log log 1 -F(x -B log(xi - C) +B log A (4.6-4)
i=1 i
simultaneously with respect to A, B and C. If we equate to zero the
partial derivatives of SSD, with respect to A, B and C, we will
produce a set of non-linear equations in the parameters which can be
solved, laboriously, by an iterative method. In addition, for this
purpose starting values for the parameters need to be obtained by
a-quick method of estimation, say, the graphical.
We instead suggest using the computing technique which we proposed for
the 3-parameter lognormal distribution in 3.12.1.
Let Vi = log log 1-F () and U1 = log (xi - C) . i
If C was known, or was fixed, to be equal to Co, say, then the points
(Ui, V1) would be coiapletely determined from the sample. Hence a
straight line, say
V= all +b (4.6-5)
could be fitted by the least squares regression technique with values
of a and b calculated from (3.7-24) and (3.7-25) of chapter 3. The
estimates, of the unknown parameters A and B would then be
B=a (4.6-6)
and � A= exp (- ä) (4.6-7)
Because the value of C is unknown our computing procedure,
described fully in section 3.12.1, is to find directly the minimum
value of SSD simultaneously with respect to C, A and B. For this
purpose we fix C, systematically, at different values and for each
168
value estimate A and B. according to the previous paragraph, and find
their corresponding value of SSD from equation (4.6-4). Hence we find
that value of C, with the required degree of accuracy, and its
corresponding parameters A and B which give the miniwarm value of
SSD. These values of A, B and C are the least squares estimates of
the Weibull parameters. Notice that for this procedure we do not
require anv starting values for the parameters. The procedure when
programmed on the computer is very fast (depending on the required
degree of accuracy) and easy to use on the interactive terminals.
4.6.2 Multinomial Maximum Likelihood Method
This method was previously considered for the lognormal distribution
in section 3.7.7. Here we will give the loglikelihood function for
the 3-parameter Weibull distribution. The computing technique to
find the estimates of the parameters, by maximizing the loglikelihood
is that described in section 3.7.7.
Let us assume that we have a sample of grouped data as defined in
section 4.6. Let pi be the probability that an observation (claim)
occurs in the interval (xi_1, xi). Adopting the notation of 4.6, we
have.
pi = W(xi; C, A, B) - W(xi_1; C, A, B)
for i=2,3, ... ,k
and
pl = W(xl; C, A, B) - {V(C; C, A, B)
Using (4.2--1) we have
cB xi_CB Pi. = exp -
(~l, -lA exp -) -A
for i. = 2,3, ... ,k 169
and xlC
B
p1 - 1-eýL A-
The likelihood function of the sample is proportional, to L where ,k ni
L=i pi
and the loglikelihood function, log L, is
k log L= nl log pi
i=1
xl -CB = nl {1 - exp
B x. -Ck C_e1 (4.6-8)
i=2 CA) ICCA )B1} i=2
The maximum likelihood estimates of the parameters are obtained by
simultaneously maximizing log L with respect to A, B and C. The
computing procedure of section 3.7.7. requires starting values for the
parameters. The least squares, or some other, estimates of A, B and C
may be used for this purpose.
4.7 Application of the Weibull Model to the AD Data
We wrote computer program P14 for estimating the parameters of the
Weibull distribution, by the least squares method, from a sample of
grouped data. This program mas run with samples of accidental damage
data as presented in tables (1.1) to (1.7). For each sample first
C=0 (i. e. the 2-parameter Weibull) was tried. The program provided
estimates of the parameters., A and B, as well as an extensive table of 2
results. The Chi-square statistic, X, was calculated by the program
170
as a measure of agreement between the fitted model and actual sample
values. A sui ry of the results is presented as part of table (4.1).
It is apparent that A has increased over time while B is, approximately 2
constant, about 0.9. Me X statistics are large for all the samples,
hence indicating a general disagreement between the model and actual
sample values. Examination of the components of the Chi-square
statistics showed that very large contributions are made by a. small
number of the intervals in the lower tail of the distribution.
For each sample, the 3-parameter distribution was then fitted by
finding the optimum value of C as described in section 4.6.1. The
results are summarized in table (4.2). It can be seen that estimates
of C, the anaunt below which claims are not made, is about 120. The
values of A and C have generally increased over time. Because the
mean and standard deviation of the Weibull distribution are functions
of all three parameters, they showed an increasing trend over time as
well. The Chi-square statistics are considerably smaller than in the
2-parameter case. However, they still indicate highly significant
differences between the model and actual sample values. Again the 2
major contributions to X values are from 2 or 3 of the intervals only.
The total expected loss statistics are relatively small as indicated
by R, the ratio of T to total actual cost of claims. Therefore, the
3-parameter Weibull distribution provides a better fit, in terms of 2
the X statistic, to the actual data.
To see how closely the least squares line fits the sample points
the computer program P15 was written to plot them. The graphs for
the accidental damage data arc presented in figures (4.3-a) and (4.3-b). 2
Despite the large values of the X statistic we observe that the
sample points lie very closely on the least squares line. This, of
171
course, was expected from the graphical test which we carried out
previously.
We next wrote computer program P16 to estimate the parameters of
the 2-parameter Weibull distribution by multinomial maximum likelihood
(IiIL) method in order to see how the results compare with those of the
least squares method. For each of the accidental damage samples the
least squares estimates of A and B were used as starting values for
the iteration process. A summary of results is presented as part of
table (4.1). We can see that the estimates of A are very nearly the
same as their corresponding values by the least squares method but the
values of B are slightly larger. The Chi-square values, however, are
considerably smaller even when the slightly smaller degrees of freedom
are allowed for. The required pooling of the intervals, in the upper 2
tail of the distribution, for the calculation of the X statistics
were different from those of the least squares method. This resulted
in the difference in degrees of freedom. The ? C2 statistics, as
before, show significant differences between the model and actual
sample values. Based on the Chi-square goodness of fit test it is
observed that the 2-parameter Weibull distribution does not provide a
satisfactory model for the distribution of accidental damage claim
amounts. 0'
Computer program P17 was written to estimate the parameters of
the 3-parameter Veibull distribution by DfiIL method. For accidental
damage samples, the least squares estimates of the parameters A, B and
C, as given in table (4.2), were used as starting values for the iteration
process. For each sample the progr= prints estimates of the parameters,
the mean and the standard deviation of the fitted distribution of
172
claim amounts and also an extensive table which, for each interval,
provides the actual, the expected and the actual minus expected
number of claims as well as the expected loss and the contribution to
the Chi-square statistic (for those intervals where the expected
number is greater than 5). The results for the accidental damage
samples are presented in tables (4.3) to (4.9). The estimates of C
are about 15 and the estimates of A are smaller than their corresponding
values by the least squares method. The values of A, the estimates of 2
A, have generally increased over time. The X statistics are smaller
than their corresponding values by the least squares method. However,
they still indicate significant differences between the model and the
actual sample values. The major contribution to the Chi-square
statistic is from few intervals only. Otherwise the agreement in most
of the intervals is satisfactory. The total expected loss statistics
are small and in every case are less than It of the total actual cost.
The estimates of B are smaller than 1 for each of the sairples. This
means that the 1eibull frequency curve resembles the exponential
curve, but with a longer tail, and has no mode. Therefore, the model
ignores the distinct mode of the histograms of the accidental damage
data.
To see how closely the frequency curve of the 3-parameter Weibull
rmdel agrees with the histogram of the sample values we modified
computer program P7 to plot them. The graphs are presented in
figures (4.4.1) to (4.4.7). In each case the exponential shape of the
curve, with no mode, is observed and the tail of the curve is seen to
be shorter than the tail of the histogram. Large deviation between
the curve and the histogram can be observed at several intervals. This
could, of course, be expected. by looking at the large values of actual
173
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minus expected frequencies in the tables. It can be noticed that
the frequency curve fits the sample histogram rather well in the
middle but not at the tail values. The agreement between the frequency
curve of the model and the histogram of the data is not satisfactory
for any of the samples.
From the above analysis we can conclude that the Weibull
distribution does not provide an adequate mdel for the distribution
of the accidental damage claim amounts.
174
4.8 The Effects cif ' Iriflatiori *ft, * the Parameters of the Weibull. Model
Following the argument of chapter 3 about the effects of inflation
on the ; arameters of a model, it is considered important to study these
effects theoretically for the Weibull distribution. We will need to
allow for such effects when predicting the future distribution of
claim amounts for a class of general insurance business in which the
Weibull yodel has been found to represent the distribution of claim
amounts.
Let us again assume that the effect of inflation is to increase
a claim of amount X to X(1 + i) over a period where i is the effective
rate of inflation for that period. If X is distributed as a 2-
parameter Weibull with distribution function V, (x; A, B) then by a
transformation of variables we can show that Z= X(1 + i) is distributed
as a Weibull W(z; Ag, B) where g=1+i.
If X has a 3-parameter Weibull distribution, W(x; C, A, B), then again
by a transformation we can show that Z= X(1 + i) is distributed as
W(z; Cg, Ag, B). Therefore inflation increases the parameters C and
A but leaves the shape parameter B unchanged. The mean and the
standard deviation of the Weibull distribution are functions of all
three parameters and hence both increase over time due to inflation.
Although it has been found that the Weibull model is not completely
adequate for the distribution of accidental damage claim amounts,
nevertheless the increasing trends of C and A can be observed from the
tables of results.
Since the 11eibull is not a very satisfactory model for our data
we do not use it to predict the distribution of claim am is during
future periods.
175
4.9 Conclusions
The 2 and 3 parameter ý`'eibull distributions were studied in this
chapter as models for the distribution of claim amounts in general
insurance and in particular for the accidental damage claim amounts.
The Weibull distribution has been regarded as a more flexible model than
the exponential distribution and hence its use in various classes of
general insurance, where the latter has been successfully applied, is
recommended. An interpretation was offered for parameter C as the
amount below which claims were not made. The multinomial maximum
likelihood method is preferred to the least squares method for
estimating the parameters of the Weibull distribution since the former
method produced smaller Chi-square statistics and hence a better
agreement between the model and actual sample values. The 3-parameter
distribution represents our accidental damage data better than the
2-parameter distribution when judged by the Chi-square goodness-of-
fit test. i-bwever, even the 3-parameter model does not satisfactorily
represent the distribution of accidental damage claim amounts. This
was indicated by the significant differences between the nadel and the
actual sample values as judged by the values of X2. The value of the
shape parameter B was found to be less than 1 which indicates that
the model has an exponential-like shape with no mode. Hence the model
ignored the distinct mode which is observed in the histograms of our
sample values. It was observed that for each sample the frequency
curve of the model had a shorter tail than the histogram of the data.
The inflation was shown to affect the location and scale parameters,
C and A respectively, but to leave the shape pur, :, eter, B, unchanged. The effect of inflation in increasing C over time is plausible as it
is expected that with inflation the amount below which clams are not
made should be increased.
176
We may note down in conclusion that the 3-parameter Iveibull
nadel was riot found to be as adequate as the 3-parameter lognormal
model in -representing the distribution of our, accidental damage claim
antunts.
4.10 Tables
177
Table (4.1)
Two-parameter Weibull model fitted to accidental damage data by two methods of estivation
Least Squares Method Mult. Max. Likelihood Method
Period of accident
A B X2 (D. F. ) A B X2 (D. F. )
73/4th quarter 143.3 0.853 224.4(24) 147.4 0.969 136.8 (21)
74/1st 146.5 0.899 182.0(22) 146.6 0.966 148.0 (21)
74/2nd " 150.6 0.914 157.1(22) 150.7 0.998 115.8 (21)
74/3rd " 163.6 0.886 235.6(24) 165.5 1.000 160.6 (22)
74/4th " 172.5 0.911 162.8(24) 173.0 0.994 122.4 (22)
75/1st 169.5 0.878 181.3(24) 176.5 1.040 87.5 (22)
75/2nd " 178.6 0.900 166.3(24)_ 1 178.4 0.988 118.4 (22)
Table (4.2)
Three-parameter h'eibull m del fitted to accidental damage data by least squares method
Period of accident
C A B 2
X (D. F. ) R%
73/4th quarter 21 108.4 0.740 100.5 (23) -0.2
74/1st " 19 114.4 0.780 83.3 (22) -0.2
74/2nd " 19 118.4 0.796 75.1 (22) -0.8
74/3rd 22 123.9 0.760 137.0 (24)_ _-0.6
74/4th " 20 136.4 0.794 75.3 (24) -0.2
75/1st " 22 129.9 0.761 102.0 (24) -0.2
75/2nd 21 139.5 0.782 74.2 (24) -0,3
i N. B. -X is the Chi-square statistic with (D. P. ) degrees of frecdcm.
R is the ratio of total expected loss statistic, T, to the total actual cost.
178
Table (4-3.1).
1! ### 3-PARAMETER WEIBULL DIS.
73/4TH QUARTER DATA
ES TIMATION BY MULTINOMI AL &'AX. LIKELI HOOD METHOD ; - C= 15.73 A= 1 22.088 f3= 0.832
MEAN= 150,349 S. D .= 162,617
-AMOUNT £ ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A-E)**2/E
1- 30 478. "482", -4. -62. 0.033 31- 60 518. 590, -72. -3276. 8.786 61- 911 461. 407. 54. 4077. 7.165 91- 120. 359. 304. 55, 5802. 9.951
121- 150 239. 234. 678. 0.107 151- 180 213. 184. 29, 4799. 4.571 181- 210 148, 147, 195. 0.007'" 211"- 240
, 102. 119. -3833. 2.429
241- 270 .
81. 96. -15. -3832. 2.344 271- 300 58. 79. -21. -5995. 5.582
"301- 330 66. ,
65, 1. 316. 0.015 331- 360 45. 54. -9. -3109. 1,500 361- 390 39.. 45. -6. -2253. 0.800 391- 420 35. 37.. -2. -611. 0 108 421- 450 34. 31. '1306. .
0.290 451- 480 20. 2.6. -6. -2793. 1.385 481- 510 29. 22. 7. 3468, 2.227
_511- 540 14. . 18. -4. -2102. 0.889 541- 570 8. 15. -7. -3888. 3.267 571- 600 9. 13. -4. -2.342'. 1,231 601- 700 29. 31. -2. -1301. 0,129 701- E300
_ 18. 18' 0. 0, 01000
801- 900 20. 11. 9. 7654, 7.364 901-1000 6. 6. 0. 0. 0.000
1001-1100 4, 4. 0, 0l I 1101-1200 4. 2. 2. 2301. 0,000
. 1201-1300 1. 1. 0. 0. 1301-1400 3. 1. 2. 2701. 1401-1500 1. 1. 0, 0. 1501-i609 0. 0. 0. 0. 1601-1700 1. 0. 1, 1650.. 1701-1800. 0. 0. 0, 0. 1801-1900 0. 0. 0. 0. 1901-2000 1, 0. 1. 1950. 2001-2100 0. 0. 0. 0. 2101-2200 0. 0. . 0, 0. 2201-2300 o. . 0. 0. 0. 2301-2400 1. 0, 1. 2350.
---------- ------------ -------- -------------- ----------- . 9.600 TCOTAL. 3045. 3043. 3651. --- -
. D. F. =22 TOTAL EXP. LOSS Y < 0.001
--------------- =. 0ý8 ,o TOTAL ACT, CnST -
179
Table (4.5,2)
3-PARAMETER. WEI©ULL DIS. #**
14/16T QUARTER DATA
ESTIMATION BY MULTINOMIAL MAX. LIKELIHOOD METHOD t- C- . 16.64 A=, 119.529 B= 0.819
MEAN= 149.887 5.. D. - 163.820
AMOUNT £ ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A-E)**2/E
1- 30 381. 384. -3. ' -46. 0.023 31- 60 428.
. 485. -57. -2593. 6.699 61- 90 351. 328.. - 23. 1736. 1.613 91- 120 334. 243. 91. 9600. 34,078
121- 150 211. 186. 25. 3387. 3.360 151- 180 133. 146. '. -13. -2151.. 158 1. 181-, 210 98. 1.16. -18. -3519. , 2.793 211- 240 82. 94. -12. ' -2706. 1.532 241- 270 54. 76. -22. -5621. 6.368 2,71- 300 52. 62. -10. -2855. 1.613 301- 330 53. 51. 2. 631. 0.078 331- 360 36. 42. -6. -2073. 0.857 361- 390 29. 35. -6. -2253. 1.029 391- 420 26. 29. -3. -1216. 0.310 421- 450 22. 25. -3, '-1306. 0.360 451- 480 22. 21. 1. -466. 0.048 481- 510 17. 17. ' 0. 0. 0.000 , 511- 540 10. 15. -5. -2627. 1.667 541- 570 19. 12. 7. 3868. 4.083 571- 600 " 4. 10. -6. -3513. 3.600 601- 700, 26. 25. 1. ' 651. 0.040 701- 800 21. ' 15. 6. 4503. 2.400 801- 900 11. 9. 2. ' 1701. 0.444 901-1000 10, 5. 5. - 4752. 5.000'
1001-1100 5. 3. 2. 2101.. 1101-1200 2. 2. 0. 0. 120 1-1300 1. 1 ., 0l 0. 1301-1400 2. 1. 1. 1350. 1401-1500 o. 1. -1. -1450. 1501-1600 0. 0.. 0. 0. 160 1-1700 0. 0. 0. 0. 17.01-1600
----------
1.
-----------
0.
--.. _------ 1.
"'- 1751. . .
_- - -- 1.125
TOTAL 2441. ' 2439. _ 2586
-- OD. 279 O. P79
--------------- .. ---- -------------- -------------------- -- ' D. F. , ýL1 TOTAL EXP. LOSS
----- _-_-------= 0.7 "w P <0.001 TnTAL ACT. Cn ST
180
Table (403D3)
*** 3-PARAMETER WEIBI$LL DIS. ***
74/2ND NUARTER DATA
ESTIMATION BY MULTINOMIAL MAX. LIKELIHOOD METHOD : - C= 15.32 A- 126.376 E3 0.862
PAEAN= 151.603 S. D. - 158.593
AMOUNT £ ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A-E)"""2/E
1- 30 351.. 354. -3. -46. 0.025 31- 60 380. 451. -71. -3230. 11.177 61- 90 382. 321. 61, " 4605. 11.592 91- 120 295. 243. 52. 5486..
. 11.128 ` 121- 150 211. 189. 22. 2981. 2.561
151- 180 . 142, 150. -8. -1324. 0,427 181- 210 114. 120. -6. -1173. 0.30 1 211- 240 101. 97. 4. 902. 0.165
"241- 270 57. 79. -22. -5621. 6.127 271- 300- 51. 64. -13. -3711. 2.641 301- 330 39. " 53. -14. -4417. 3.698 331- 360 36. 43. -7. -2418. 1,140 361- 390" 25. " 36. . -11. -4130. 3.361 391- 420 24. 30. -6. -2433. 1.200 421-. 450 27. - 25.
. 2. 871. 11.160'
451- 480 18. 21. -3. -1396. 0.429 481- 510 21. . 17. 4. 1982. 0,941 511- 540 17. 14.
_3. 1576. 0.643
541- 570 12. 12. 0. 0. 0.. 000
"571- 600 '11. 100" 1. 585. 0.100 601- 700 30. 24. 6. . 3903. 1.500 701- 800 13. 13. 0. 0. 0.000 801- 900 11. Be 3. 2551. 1025 90-1-1000 4. 59 -1. -951. 0.200
'1001-1100 7. 3. 4. 4202.
1101-1200 0. 1 2. -2. " -2301. 1201-1300 1. 1. 0. 00 1301-1400 1.. 1. U. 01 1401-1500 0.
" 0. 0. 0f
1501-1600 1. 0. 11. 1550. 1601-1700 0. 0. 0. 0,
. 1'101-1800 0. 0. 0. 0. 11301-1900 1. 0. 1. 1850. 2.7°6 -------------------------------- --------------------------------
TOTAL -----------
2383. ----------
2386. ----- ------ - ----------
-106. -----------
63,124 -------- -
D. F. =21 TOTAL EXP. LOSS r
-----------------a -010 % , <0.001
TOTAL ACT. COST
4
181 %
Table ('1,3.4)
11 . .. *** 3-PARAUETER WETf3UL. L DIS, , ***
74/3RD QUARTER DATA I
ESTIMATION BY MULTINOMIAL MAX. LIKELIHOOD METHOD : - C- 16.07 A= 139.472 B= 0.867
MEAN= 166.012 S. D. = 173.566
AMOUNT £ ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A-E) **2 /E
1- 30 362. 366. -4. -62. 0.044 31- 60 427. 5.02. -75. -3412. 11.205 61- 90 383. 365. 18. 1359. 0.88A 91- 120 356. 281. 75". 7912. - 20.018
121- 150 283. 223. 60. 8130. 16.143 151, - 180 194. 1791. 15. 2482. 1.257 181- 210 137. 146. -9. --1759. 0.555 211- 240 97. 119. - -22. -4961. 4.067 241- 270 . 86. 981 -12. -3066. 1.469 271- 300 .
71. 82. -1.1, -3140. 1.476 301- 330 64. 68,, -4. -1262. 0.235 331- 360 45. 57. -12, -41.46. 2.526 361- 390 44. 48. -4. -1502. 0.333 391- 420 - 25. 40. " --15. -6082. 5�625 421- 450 26. 34. -80 -3484. 1,882 451- 480 22. 28. -6. -2793. . 1.286 481- 510 25. 24. 1., 496. 0.042 511-. 540 14. 20. -6. -3153. _ 1.800 541- 570 14. 17. ' -3. -1'666. 0.529 571- 600 17. 15.
. 2. 1171. 0,267 601- 700 32. 35. -3. -1951, 0.257 "701- 800 34. . 21. 13. -9756. 8.048 801- 900 17. 12. 5. 4252. 2.083 901-1000 4. 8. -4. -3802. 2.000
1001-1100 9. 5. 4. 4202. 3.200 1101-1200 4. 3. 1. 1150. 1201-1300 0. 2. -2. -2501. 1301-1400 1. ,
1. 0. 001 1401-1500 3. 1. 2. 2901. 1501-1600 0. 0. " 0. 0. 1601-1700 1'. 01 1. " 1650. 1701-1800 0". 0. 0, 0. 1801-1900 0. 0. 0. 0. 1901-2000 0. 0. 0. 01 2001-2100 0. 0. 0. 01 2101-2200 1. 0"
. 1" 2150.
2201-2300 0. 0. 0. 0. 2301-2400 0. 0. 0, 0. 2401-2500 0. 0. _
0, 0. . 2501-2600 -
. 1.
- 0. 1. 2550. 2: 286-
TOTAL
----------- 2799.
----------- 2800.
----------- ----------- 141gß'--
--------
-----__ý__ 92.722
----------- D. 'E. ^2
TOTAL EXP. LOSS --------- 0.3 ° P <0.001
TOTAL ACT. COST
182
Table `4-3-5)
*# 3-PARAMETER WEIBULL DIS. '***
74 /4TH QUARTER DATA
ES TIMATION BY MULTINOMIAL AJAX. LIKELIHOOD METHOD : - C= 15.15 A= 148 . 275 E= 0.871
" MEAN= 174.111 S. O. = 183.102
"AMOUNT £ ACTUAL EXPECTED ACTUAL- 'EXPECTED CL. NO. CL. NO, EXPECTED LOSS. (A-E)**2/E
1- 3.0 394. 397. -3. -46. 0.023 31- 60 452. 522. -70. -3185. 9.387 61- 90 426. 386. 40. 3020. 4.145 9.1-"120 348. 302; 46. 4853. , 7.007
121- 150 272. 241. 31. 4200. 3.988 151--'180 219. 196. 23. 3806. 2.699 181-210 154. 161. -7. -1368. 0.304 211- 240 124. 133. -9. -2029. 0.609 241.270
. 105. 111. -6. -1533. 0.324
271- 300 78. " 93. -15. -4282. 2.419 301- 330 75. 78. --3. ' -947. 0.115 331- 360 58. 65. -7. -2418. 0.754 361- 390 55. 55. 0. ' 0. 0.000 391--420 29. 47. -18. -7299. 6. x94 421-' 450 -43. 40. 3. 1306. 0.225 451 480 24. 34. -10. -4655. 2.941 48'1- 510 22. 29. -7. -3468. 1.690 511-"540 24. 25. -1. . -525. 0.040 541=- 570 19. 21. - -2.
-1111, 0.190 571='600 14. 1A. -4. -2342. 0.889 601- 700 42. 43. 1. -651. 0.023 701- 800 28. 26., 2. -" 1501. 0.154 801- 900 20. 16. 4. 3402. 1.. 000 901-1000 17. 10. 7. 6653. 4.900
1001=-1100 a. 6. -2. 2101. 0.667 1101-1200 5. 4. 1. 1150; F 0.290 1201-1300 1.
" 2. -1. -1250.
1301-1400 5. 2. 3. 4051. 1401-1500 0. 1. -1. -1450. 1501="1600 1. 1. - 0. 01 1601-1700 0. 0. 0. 0. 1701-1800 0. 0. 0. 0. 1801-1900 0" 0. 0. 0. 1901'-2000 0. 0. 0. 01 2001-2100 1.. 0. 1. 2050. 2101=2200' 0. 0. 0. 0. 2201-2300 0.
, 0. 0. 0.
2301-2400.
------------- ----------- ------------ 1.
-------- 2350�
--------- 1. . 500
TOTAL` 3064. _ 3065. ----- 1, ------- _ -------"--------------.. - ---» .. -------- , -----i--.. --------....
--- 5 L
-ý.. ý
--
ý
TOTAL EXP. LOSS D. F. s'
---------------= 0.4 % P Co. (, )al TOTAL ACT. COST
183
75/ 7ST N. "AfTE° L, "-, Y FM LI// a-ul-a. (4.3.6/
ESTIMATION BY MULTINOMIAL MAX. LIKELIHOOD METHOD ; C- 14.89 .
A= 149.659 E3= 0.890
MEAtJ= 173.325 ;. o. = 178.362
A(ACUNT £ ACTUAL. EXPECTED ACTUAL-- EXPECTED CL'. NO. CL. NO. EXPECTED LOSS (A-E)* 2/E 1- 30 324. 326. -2,. -31. f'i. 012 31- 60 387. 439. -52, -2366. 6.159 61- 90 345. 330. 15. 1132. 0.682 91- 120 289. 260. 29. 3059. 3.235 121- 150 253. 209. 44, - 5962, 9.263.
151- 180 1(i7. 170. 17. 2813. " 1,700
181- 210 138. 1461. -2, -391. , 0.029 211° 240 114. i16.
-2. -451. 0.034 2411 - 270 93. 96. -3. -767. 0.094 271- 300 67. 80. -13. -3711. 2.112
, 301-, 330 63. 67. -4. -1262. 0.239 331- 360 449 56, -12, * -4146. 2.571 361- 390 44. 47. -3. -1126. 0.191 391- 420 35. 40. -5. -2027. 0.625
'421- 450 25. 34. -9, -3919. 2,382 `451-. 4130' 26. 29. -3. ý-1396. 0.310 -. 481- 510
. 18. 24. -6, -2973. 1.500
511'- 540 18, 21. -3. -1576. 0.429 541-570 22, 18. 4. 2222. 01889 -57.1- 600. 17. 15. 2. 1171. 0.267
"601-'700 390 36. '
3. 1951. 0.250 701- -800 19. 21. -2. -1501. 0.190 801-- 900 18. 13:. 5. . 4252. ' 1.923 901-1000 12. . 8" 4. 3602, 2.000
1Q01-1100 3, 5. -2. -2101. 1101-1200 1. 3. -2.. -2301'.
aý 00 1201'-1300 1, 2. -1. -1250. 1301-1400 00 " 1. -1, -1350. 1401-1500 1. 1. U. U. 1501-160: ) 0" 0. 01 01 1601. -1700 1. 0. 1. . 1650.
:, 1701-1800 0. 0. 0. 00 1801-1900 0. 0. 01 0.
, 1901-2000 0. 0. 00 0. 2001-2100 1. 0. 1. 20501, 2101-2200 1. 0. 1. 2150. 2201-2300 o, 0. . 0. 00 2301--2400 01 0" 0.
% 00
2401-2500 0. 0, [): p. 2501-2600 0.. 0. 0. 00 2601-2700 0. p. a. 0. 2701-2800 0. . o. 0. 0 28012900 U. 0. * 01 0..
'2901-3000 0. . 09 0. 0. 3001-3100 0. 0. 01 0. 3101-3200 U. U. . U.
. p. 3201-3300. 0. 01 0.. 3301-3400 0. - 0. 01 0, 3401-3500 0, 0, 0, 0. 3501.3600 1, 01 1" 3550 ----- ----.. w _.. w-_--_-s_ ---w _. ýw-.. w -.. wM------- ----------- - _0.
c, ()G TOTAL 2607L. 2607, 1 120 . ,. n <p-)
,,, --------------------------------------------------
TOTAL EXP. LOSS D. F. G?.
---- .... _--_-_. -- 0,2 9 .°
4t, a 2
TOTAL ACT,. C: 0. )ß
(x.. 3.7)
'3-PARAMETER 1r+E'IBULL DIS.
75/2ND QUARTER DATA -
ES TIMATION BY MULTINOMIAL MAX. LIKELIHOOD METHOD ; - C= 16.52 A= 150 , 912 B= 0.856
MEAN= 179.934 "S. D. = 191.534
AMOUNT £ ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. ' NO. EXPECTED LOSS (A-E) *2 /E
1- 30 302. 305. -3. ' -46. ' 0.030 31- 60 374. 428. -54. -2457. 6.813 61- 90 332. 312. 20. 1510. 1.. 282 91- 1.20 277. 243. 34. 3587. 4.757
121- 150 235. 195. 40. 5420. 8.205 151- 180 187. 159. 28. 4634. . 4.931 181= 210 122. 131. -9. -1759. 0.618 "
' 211- 240 1.10. 108, 2. 451. 0.037" 241- 270 80. 90. -10. -2555. 1.111 271- 300 72. 76. -4. -1142. 0.211 301- 330 47. 64. -17. -5363. 4.516 331- 360 39. 54. -15. -5182. 4.167 361- 390 40. 46. -6. " -2253. 0.783 391- 42.0 38. 39. -1. . -406. 0.026 421- 450 29. 33. -4. -1742. 0.485 451- 480. 21. 29. -8. -3724. 2.207 481- 510 30. 24. 6. 2973. 1.500 511- 540 19. 21. . -2. -1051. 0.190 541- 570 17. 18. -1. -555. 0.056 571- 600 11. 16. -5. -2927. 1.562 601- 700 36. 38.
. -2. -1301; 0.105 701- 800 22. 24. -2., -1501. 0.167 801- 900 22. 15. 7. 5953. 3.267 901-1000 11. 9. 2.
. 190,1. 0.444 1001-1100 4. 6. -2. " -2101. 0.667 1101-1200 3.
" 4. -1. --1150. " 0,250"
1201-1300 3. 2. 1. 1250. 1301-1400 6. 2. 4. 5402. 1401-1500 2. 1. 1. 1450. 1501-1600 1. 1. 0. . 0. 1601-1700 0. 0. 0.,
, 0. 1701-1800 1. 0. "" " 1.
.- 1751.
1801-1900 .
J. 0. 0. 0. 1901-2000 1. 0. 1. 1950. 2001-2100 0. 0. 0. 0. 2101-2200 0. , 0. 0.. 0. 2201-2300
------- --
1. -----------
0. ------------
1. ----- -
2250. 13.! ()0 --
TOTAL
----------- 2495.
----------- 2493.
------------
- ----
----------
---------- -"
3266. -------
61.065
- ----------- D. F,
------- =23 TOTAL EXP. LOSS
---------------- 0.7 % P<0.001
TOTAL ACT. COST
185
CHAPTER 5
The Liverse Gaussian Distribution
5.1 Introduction
This is a uninodal and positively skewed distribution which in
recent years has been shown to represent the distribution of random
outcomes of various phenomena. The name inverse Gaussian (or inverse
normal) was proposed by Tweedie (1947) who noticed the inverse
relationship between the cumulant generating function of this distribution
and that of the normal distribution. Tweedie (1957) investigated the
statistical properties of this distribution and indicated analogies
between its statistical analysis and that of the normal distribution.
In recent years this distribution has been studied by many authors.
Johnson and Kotz (]. 970) provide a bibliography, and Chhikara and Folks
(1978) give a review of the development since 1915 of the inverse
Gaussian distribution and of the statistical methods based upon it.
The frequency curve of the inverse Gaussian distribution is
similar in shape to that of the other skewed distributions like the
lognormal, the gamma or the Weibull. It is, therefore, important to
consider the application of this distribution in the fields where one,
or some, of the others have been successfully used as a model. Chhikara
and Folks (1978) give examples of sets of data equally well fitted by
the lognormal and inverse Gaussian distributions. They point out that
when more than one distribution fits a set of data equally well and
there is no evidence (based on the physical considerations of the
problem) in favour of a particular nadel then it is best to use that
which is more convenient to work with. An advantage of the inverse
Gaussian distribution over the lognormal is that its statistical
properties and inference procedures are well developed for small sample
situations.
If)t O
Applications of the inverse Gaussian distribution in various
fields have been reported by many authors. It has been successfully
used in life-testing and reliability studies by Chhikara and Folks
(1977). Hasofer (1964) considered it as a model for emptiness of a dari.
Lancaster (1972) used it as a model for the duration of strikes. We
have not cone across the use of this distribution in insurance, and
it is therefore considered important to study it as a model for the
distribution of claim amounts.
In the literature, only the 2-parameter inverse Gaussian distribution
his been considered. We will, however, introduce a location
parameter into the model and will name it the 3-parameter inverse
Gaussian distribution. Professor J. L. Folks, in a personal communication
with the author (1978), acknowledges that this is an original idea arcs
that this form of the distribution (i. e. the 3-parameter) has not been
previously considered. We will later show that the 3-parameter
distribution provides a better model for the accidental damage data.
In this chapter we will initially define the 2 and 3 parameter
inverse Gaussian distributions and will mention some of their
properties. The problem of estimation from grouped data is next
considered. In section 5.5 the inverse Gaussian models will be fitted
to our seven samples of accidental damage data and the results will
be discussed. The effect of inflation on the parameters of the model
will be studied in section 5.6 and, for the accidental damage data,
the distribution of claim amounts during a future period will be
derived and compared with the actual data. The findings of this
chapter will be sunrarized in section 5.7. The tables will be presented
in section 5.8.
5.2 Definition
A random variable X is said to have a (2-parameter) inverse
187
Gaussian distribution if its probability density function (p. d. f. ),
denoted by f IG (x; u , a) , is of the form
2 IG (x; tý ýxý _ (X/2'rx )
1/2 Pxp (_x(x_i) /2u2X) 'x>0
=0 otherwise
where }i and X are two independent and positive parameters. This
distribution is thus a member of the exponential family. A is a shape
parameter while p is only partially interpretable as a location
parameter.
Shuster (1968) has shown that the inverse Gaussian distribution function,
denoted by IG(x; u, a), can be expressed in terms of the standard
no-mal distribution function, N(z; 0,1), as :
IG(X; u, a) = N(zl; 0,1) + eta/u rß(72; 0,1) (5.2-2)
where Zl = .
(A/x): V2
(-1 + x/u)
z2 = -(x/x) (1 + x/u) I't
1e will show later in this chapter that the 2-parameter inverse
Gaussian distribution does not provide a satisfactory model for the
accidental damage claim amounts. This fact, together with the"concepts
of 3-para eter lognormal and SWleibull distributions, led us to consider
a 3-parameter inverse Gaussian distribution where one of the parameters
is a location (threshold) parameter below which the values of the
random variable X are not realized. Hence we introduce the unk noun
location parameter c into the nwdel and assume that X-c, rather
than X, is distributed as in (5.2-1).
We say that a random variable X has a 3-parameter inverse Gaussian
distribution if its p. d. f., denoted by fIG(x; c, u, X), is of the form
2
IG(x, [a/21r (x-C)3) exp[-x (x-c-u)` /2 i (x-C)] ;X>c =0U Lll--Il iSC
(J"2-�)
88 1
The transformation x- x-c is only a translation along he x-axis.
Therefore, the distribution function of X(i. e. the 3-parameter inverse
Gaussian random variable) is obtained from (5.2-2) by replacing x by
X-C.
In our research into the literature on the inverse Gaussian
distribution, we have not come across the 3-parameter distribution of
the form (5.2-3). In fact, as we mentioned earlier, Professor Folks
in a personal communication with the author confirms that the idea of
a 3-parameter inverse Gaussian distribution, with one of the parameters
being an unknown location parameter, has not been considered before.
This new form suggests, as an area for research, the study of th3
statistical properties and the estimation and inference problems for
the 3-parameter distribution. We shall deal with these problems in
this chapter as far as is relevant to the present work. It will be
shown that the 3-parameter distribution provides an adequate model for
our accidental damage claim amounts.
5.3 Properties of the inverse Gaussian Distribution
In this section, we first outline briefly sorge properties of
the 2-parameter distribution with p. d. f. of the form given by (5.2-1).
for more details and proofs reference should be made to Johnson and
Kotz (1970) or TNAreedie (1957).
Moments of all orders exist and in particular the rth positive
integral moment of X about 0 is given by Tweedie (1957) as :
1 I E (Xr) =r S=0
In particular E (X) 2
and E (X )
hence var(X)
r-l+s )s s! (r-1-s)! 2x
(5.3-2) 23 -1 +ua (5.3-3)
3 ä (5.3-4)
18
Also I: (X3) = 11 3+
311 x-1+ 31_a ` (5.3-5)
4 ý4 5 _1 6 -2 73 and E(X =p+ 611 X+ 1511 X+]. Su X (5.3-6)
Parameter p is, therefore, the population mean and is a measure of
location. This model, unlike for instance the lognormal model, possesses
the useful characteristic that one of its parameters is the population
mean. 2 The coefficient of variation for this distribution is ý-
2 where
4, >0 (5.3-7)
The coefficients of skewness and kurtosis, V-O1 and g2 respectively, are
(5.3-8)
02 =3+1=3+ 15q-i (5.3-9)
The shape of the distribution depends on + only and, since ý>0, its
frequency curve is skewed to the right and more peaked than the normal
frequency curve. The distribution becomes more and more nearly normal
when ý is increased.
The distribution has a single mode which is located at
or
r2y + )2 Xmode -u{ (1 _} (5.3-10)
` 4x 2
9z 31 + 44,2) 2ý (5.3-11) xirýode -u (1
The mode of the distribution is in general located to the left of
its mean because
Xnn de t u
but when ý is increased to infinity x, rode wiil converge to 11 . It is
possible to express the inverse Gaussian p. d. f. in terms of the pair
of parameters (u, ý) or (x, ý) , but the form give, by (5.2-1) is
T£0
generally considered as the standard form and is most usually used
in the literature. The one parameter form which is obtained by putting
p=1 in (S. 2-1) is knom as the standard Wald distribution.
The properties of th4 3-parameter inverse Gaussian distribution,
whose p. d. f. was given by (5.2-3), can be deduced from those of the
2-parameter. The transformation x -} x-c is one of translation along
the x-axis only and, therefore, the measures of location will be
increased by c while the measures of dispersion and the moments about
the mean will be unchanged. Hence, for the 3-parameter distribution
we will have :
E (X) =c+u (5.3-12)
var(X) = l' (5.3-13)
=r2 112 1
Xinode c+ u{ (1 + 9u2)
- 2ý } (5.3-14) tt 4ý JJ
3/2
and Coeff. of variation =u (5.3-15) [X(c )] 2
which is a function of all three parameters.
5.4 Estimation of the Parameters from Grouped Data
The problem of estimation from a sample of data containing the
values of individual observations has been dealt with by Schrodinger
and his maximum likelihood estimators are reported in Chikkara and
Folks (1978). Here we concentrate on the estimation from a sample
of grouped data. when all the parameters are assumed unknown.
Let us assume that we have a sample of grouped data where n independent
random observations on a random variable X (in our case the claim
amount) have been grouped according to their size into k nutually
exclusive intervals. Further, let n1 be the niunber of observations
(claims) in the class interval (xi_l, xi), for i=1,2, ... ,k,
191
and such that
k n= n1
i=1
Also let us assume that X has a 2-parameter inverse Gaussian
distribution of the form (5.2-1). The method of moments can be used
to estimate the parameters. This involves putting as many sample
moments as there are unknown parameters equal to their corresponding
population values and solving the resulting equations for the unknown
parameters. For the 2-parameter case the sample mean and coefficient
of variation may be used. The sample mean is the moment estimator of
and, when the values of individual observations are known, is
equal to its maximum likelihood estimator (see Johnson and kotz (1970)).
Therefore the sample mean is an unbiased and efficient estimator of u.
In calculating the sample moments from grouped data the usual assumption
about the concentration of all the observations in each interval at
the mid-point of that interval may be made. In the 3-parameter
case, as well as the sample mean and coefficient of variation, the
third central moment, which is independent of c, should be used to
yield sufficient equations in the unknown parameters.
It was observed in the previous chapters that the method of
multinomial maximum likelihood yielded the best fits of the models
to actual data. Here we consider this method for the 3-parameter
inverse Gaussian. The 2-parameter case can be similarly dealt with by
putting c=0 in the following exposition.
Let us assume that the have a sample of grouped data, as defined earlier,
and that random variable X has a 3-parameter inverse Gaussian distribution
with p. d. f. of the form (5.2-3) and distribution function given by :
(see section 5.2)
IG(x; c, u, X) ° N(: 1,0,1) + e2ý/u N(z2; 0,1) for x>c
=0 otherwise (5.4-1)
192
where zi = (a/ (x - c)) 2 (-1 + (x - c) /u )
and 1 z2 c))
/ (1 + (x - c)/
Let pi be the probability that an observation (claim, in our case)
occurs in the interval (xi-1, xi) , for i=1,2, a. ,k, where
Pi = IG(xi ; c, u, X) - IG(xi-1; c, p, X) (5.4-2)
which can be expressed in terms of the standard normal distribution
function by using (5.4-1). The likelihood of the s . ple will be
proportional to L where k
ni L=
i=1 pi
The loglikelihood equation, log L, can thus be expressed as
k
log L= nl log pi i=1
The rnaximun likelihood estimates are the set of parameter values
(c, u, Jý ) which maximizes log L. The iterative computing technique
which was described in section 3.7.7 of chapter 3 can be used to AA
maximize log L and yield the set (c, u, a). This technique requires
initial values for the parameters in order to start the iteration
procedure. The estimates of the parameters obtained by the method
of mments can be used for this purpose.
5.5-Application of the Inverse Gaussian Model to AD Data
We wrote computer program P18 to estimate the paraieters of the
2-parameter inverse Gaussian distribution by the multinormal maximum
likelihood , (ASS; [. ) , method. This program was run with the samples
of accidental danage data which were presented in tables (1.1) to (1.7).
193
For each sample the estimates of the parameters u and were found
by the method of moments. The statistics given at the bottom of each
of the tables (1.1) to (1.7) made the above task very easy. The
moment estimates were supplied to program P18 as starting values for
the iteration process. The program produced an extensive table of
results for each sample. These are summarized in table (5.1). The
estimate of u in each case is practically equal to the sample mean
(refer to tables (1.1) to (1.7)). The values of u and A show a generally
increasing trend over time. The ratio of the total expected loss
statistic, T, to the total actual cost is small in each case. This
indicates an overall agreement between the model and actual data. The
Chi-square statistics are, however, large and indicate significant
differences between the model and actual sample values. The examination
of the components of this statistic, for each sample, showed that 2
very large contributions are made to the value of X by only one or
two intervals. in the lower tail of the distribution. For example the 2
first and second intervals account for some 60% of the X values.
Therefore, the agreement between the nodel and actual sample values is
reasonable except in the lower tail. In terms of the Chi-square
statistic the 2-parameter lognormal model provided a better fit to the
actual sample values than the present model has.
The frequency curve of the inverse Gaussian distribution is very
similar to that of the lognormal distribution. To improve the fit of
the model to the actual data, we considered the idea of a 3-parameter
inverse Gaussian distribution similar to the idea of a 3-parameter log-
normal distribution. We, therefore, assumed that X+c, and not X, is
distributed as an inverse Gaussian distribution, with parameters u and
x, where c is an unk noun parameter which can be interpreted as the
voluntary excess on the policy. This assumption simply means that the
total amount of claim, X+c, (which consists of the amount X paid by
194
the insurance company and the amount c borne by the policy holder)
has an inverse Gaussian distribution.. The remarks we made about
parameter c of the lognormal model, in section 3.10, are valid here
as well and it is useful to refer back to then at this stage.
We, therefore, assume that c is fixed for all policies but that its
value is unknown, and proceed to estimate it, along with u and x from
the data. The computer program P19 estimates the 3 parameters from
a sample of grouped data and prints an extensive table of results.
For the accidental damage data of tables (1.1) to (1.7) the results are
presented in tables (5.2.1) to (5.2.7).
The estimates of c show an increase over time and are generally
greater than their corresponding values for the 3-parameter lognormal.
model. The mean claim amount of the model is, in each case, practically
equal to the sample mean. The standard deviations are also close to
their corresponding actual values and in each case are smaller than
those obtained by the 3-parameter lognormal model. The Chi-square
statistics are smaller than in the 2-parameter case. and (except in
two or three cases where large contributions from one or two cells 2
have resulted in large values for X) do not indicate any significant
differences between the model and the actual sample values. The Chi-
square statistics are practically equal to their corresponding values
for the 3-parameter lognormal model. The total expected loss statistics,
T, are also small and are at most 1% of the total actual cost. This
indicates a general agreement between the model and the sample values.
To see how closely the frequency curve of the model agrees with
the histogram of the sample, we wrote computer program P20 to plot
them. The ML estimates of c, u and X, given in tables (5.2.1) to
(5.2.7), were used to plot the frequency curve of the fitted model for
each sample. The graphs are presented in figures (5.1.1) to (5.1.7).
We observe an overall agreement be , Teen the curve and the histogram
for each sample. In particular, the cunfe has portrayed the distinct
195
.:, , aý frýýiluýýnr.; ý
? 3.
>-, rý >. s .v
. ýý{ý /1ýý llýTli
Figure (5.1.1)
M ^! " 73'iTH DUARTF. R DATA AND 3-PAR.
3S1AN P. G. F. WITH (1ULTU. MAX. PARAMMEf: s
MEW-1604.0 LAMDAL16ü. 5
v-r-77ý1 .
I'--- r. --,
-12.10 , (Alo
ICI ýTf1; C. ýM ^f' 73'ITIi QU\RT(R CATA /.! %1r)
I14V. GAU3$1A14 ('. r. F . {: 1 fli MULTN. M1XL. f Ai:, 1C1f: "f(fi5
I; v. o i ý, 01 wort 1 . 400
2d' I r. ` , u49 l
22.
2- 1J.
IC
'7.
16.
13.
12
11.
6
5.
` 3.
2.
CA- 3 30 6Eý 9sß 120 1
X12-1 5.00
ireouency
4.00
f3. Jul
2.5E
2. Oa.
Figure (5,1.2)
M OF. 7.1ilST CUARTE. R CATA AND 3-PAR.
ISSIAN P. U. F. WITZ; MULTN. MOL. PAPAME"TF: RS
ME. W=161.6 LAM')ALl 3.1
n^
2100 22. ̂ "( 29: 0
H1ST(X; RAM OF 7-l'lST OU'F T1: CA JA nay' ;ý "ý nR . INV. GAU3SIAN f . U. r. WITH MULTN. MNXL. ! 'AF: ^ili: TUk,
1.2.,. Ij It iv-i
Fr 6
` 23
22 2t.
20
19. iG
17.
16.
13 12.
. tt 12
Figure 5.1.3)
{,:; rO., PAi1 (`f 71 ,? NP) , )I'Ar"TF P ! '. A At i-,, '3 "PAP.
! NV. I A, 1 P. CD. r . 'iIT', MUL'1Ii. tt"/I.. c%/P4M(. 1fPS i
Lu 19 .0 11(; w'zj; 70 .? LA! ': Aý 1 f; 2 . 81
act G© S%) 12L1' i: ý© ; E£1 213 2a© 27C' 200 33c 31'1 3ý0 92,1 lt O -IE; 510 510 57: 1 Eßt)
Frequency hISTD3PAT1 OF 74'2NO CUAF. TE'. R CAA Auf 3-PAR.
INV. GAUSSIAN P. D. r-. WITH MULTN. MaXL. PAPAMf_TE: PS
C=19.0 ME: W=27, .7 LA. -", Aw162. G
ii`li tl. iý_ý... i:: {'1ý . _.
ý1' r-ý: ýi
-. ý1ý ýr. ý41ý
.. 'ý1ý"
. `. i... 'llýll "Ltyr
t
t
E
ýý i
c
}
s a
r
a
t
4
X1.,. -1 5 . 'E+ý t
` 1.50
4. ee
3 .M
I 3. ý©
2.
2.
1.
1.
Figure (5.1.4
M DP 71/3RD DUARTE. k DATA AND 3""PAR.
JSSIAN P. D. F. WITI4 MULTN. M+1XL. PAiPAMOUPS
MEWs166.7 LAt1DA=193.9
. �-�-. _--_ --. -----. --------�
; tu' 2i'. 'l : 'it: (. f: PJ ü: 1i! irißi3
Fiore (5.1.5) 'l' .
x»-1 5. ri
3. ßs
2. J(
2. kߣ
1. ýk
1 . 0C
.1 rR
!ý {1 l:. a9 1üi3f7 i: {: ý i'ýF)t) Sfý(, t1 iY F"t1 ? (i(1F1 ?: i! Fý : r"iN(9
4 .'
2.
,. t
17
1S
1^
13
12.
111
a.
x12-1 5. ro.
F
i. 5a
1. ee
3. Eh
3. ý0
2. `M
2. (C.
" 1. Sß.
"Hfl - r c; C
Pirure (5.1.6)
HISTOGRAM OF 75/1ST QWRTC. R DATA AND 3-OAR.
1NV. GAU: SIAN P. D. F. WITH MULTN. MM. PARArr. TURS
Cu2G. 0 MEW-159.2 LAMOAL233.7
It'll NX. 7 COW I jfM 0 2. "fl,
J', Flgure (5.1.7)
I
x13-ý s. ea
4
9 . CC.
:?. z. e.
2., ß.
IMO
ISTCCPAr OF 75-12ND CJARTtR DATA AND 3-PAP..
NV. GAUSSIAN P. D. F. WITH MULTN. MARL. PAPAMEUPS
=21.5 MEW=231.2 LAMDAL233.1
t. l. t NCO i: 4. Cl
mode and the long tail of the histogram very well. The curves look
remarkably similar to those for the 3-parameter lognormal distribution
given in figures (3.9.1) to (3.9.7). However, the latter curves have
a slightly fatter and longer tail.
The above analysis shows that the 3-parameter inverse Gaussian
distribution is an appropriate model for the accidental damage claim
amounts, This model produces results very similar to those of the
3-parameter lognormal distribution.
196
5.6 Prediction of the Claim amount Distribution
The importance of predicting; the future, cost of claims was
mentioned in section 3.9 (chapter 3). Here, we will adopt techniquez,
similar to those of that section for predicting the distribution of
claim amounts, during any future period, when the underlying model is
the inverse Gaussian distribution. To save us repeating some of the
arguments of that section, we suggest referring back to it at this stage.
5.6.1 The Effects of Inflation on the Parameters of the Model
Let us again assume that the effect of inflation on a claim of
amount X is to increase it, over a certain period, to U= X(i + i)
where i is the rate of inflation according to some appropriate index
for that period. If X is assumed to be distributed as the two-
parameter inverse Gaussian, IG(x; u, X) then, by a transformation of
variables, we can show that U= X(1 + i) will be distributed as
IG(u; ru, ra) where r=1+i. If X is assumed to have the 3-parameter
inverse Gaussian distribution IG(x; c, u, X) then similarly it can be
shown that is distributed as IG(u; rc, ru, rA). Therefore when
the distribution of X is known we can modify its parameters to obtain
the distribution of claim amounts in a future period. The foregoing
shows that inflation increases all the parameters of this model over
time. This is reflected in the estimates of the parameters from
different samples presented in table (5.1), for the 2-parameter
distribution, and in tables (5.2.1) to (5.2.7) for the 3-parameter case.
The data analysis of section 5.5 showed that the 2-parameter
inverse Gaussian distribution is not an adequate model for the accidental
damage claim amounts. Therefore, we will not consider this model for
prediction purposes. The 3-parameter distribution, however, was shown
to be a satisfactory model and hence we suggest using the technique
of section 3.9.2 for predicting the distribution of accidental damage
197
claim amounts during a future period. In chapter 3 it was shoi: rn tý. at
the appropriate index for calculating the rate of inflation for
accidental damage claims is the General Index of Retail Prices. We will
use the same index here again.
5.6.2 Prediction for the AD Data
To test the prediction technique on the Accidental Damage (AD)
data we wrote computer program P21 for the 3-parameter inverse Gaussian
model. For reasons presented in section 3.9.2 we used the I%HL estimates
of the parameters of the distributions for 73/4th, 74/1st and 74/2nd
quarters to predict the distributions of claim cmounts in 74/4th,
75/1st and 75/2nd quarters respectively. The rate of inflation in each
case was calculated from the General Index of Retail Prices as described
in section 3.9.2. The results for the AD data are presented in tables
(5.3.1) to (5.3.3). These are in the form of extensive tables which
allow comparisons between the actual and predicted distributions.
In all three cases the Chi-square statistics are relatively small,
considering the number of degrees of freedom, and do not indicate any
significant differences between the predicted distributions and the
actual sample values. The total expected loss statistics are small,
and show anover-prediction of the total cost, by at most 3.9%, which
is more acceptable than any under-prediction. The Kolmogorov-Smirnov
statistics were calculated from the tables and in each case were
compared with the significance points of table (2.1). They showed that
the differences between the actual and predicted distributions were
not significant for 74/4th and 75/ist quarters but were almost significant
for 75/2nd quarter.
Our prediction technique, therefore, provides a satisfactory mean:,
of predicting the future distribution of claim amoimts when the rnder"-
lying model is the 3-parameter inverse Caussien distribution. For the
198
AD data the values of the goodness-of-fit test statistics are verg
similar to those obtained from the 3-Parameter lognormal distributions
in tables (3.45) to (3.47). This is only to be expected since, as
we mentioned earlier, these two nodels are very similar to each other.
The lognomal model, however, resulted in smaller values for the total
expected loss statistics and hence overpredicted the total actual
cost by a smaller amount.
5.7 Conclusions
In this chapter the 2-parameter inverse Gaussian distribution was
initially defined and then modified to a three parameter distribution
by the introduction of a location parameter. The properties of these
distributions were considered and their problems of parameter estimation
from grouped data were dealt with. The multinomial maximum likelihood
method was suggested for these purposes. The two and three parameter
distributions. were next considered as models for the distribution of
accidental damage claim amounts. It was shown that the 2-parameter
distribution is not an adequate model, while our proposed 3-pararmeter
distribution provides a very satisfactory model which is as good as
the 3-parameter lognormal. The effects of inflation on the parameters
of the model were studied, and the future distribution of claim amounts
was predicted. It was shown that our prediction technique produces
satisfactory results when the underlying model is the 3-parameter inverse
Gaussian distribution.
The similar features of the 3-parameter inverse Gaussian and
lognormal models were pointed out at various stages. Although the latter
distribution has a slightly longer tail, and produced a Slightly better
fit to the data, either of the two models may be satisfactorily used.
The ATIL method can be used for estimation of parameters in both cases
with equal labour. The distribution functions of both yodels are
199
expressed in terns of the standard normal distribution function. It
is, however, possible to test if a given sample of data is from the
lognormal distribution, while similar tests do not exist for the
inverse Gaussian model. This is an advantage of the former model
since, if the test indicates that the sample is not very likely to be
from the distribution under consideration, then further analysis can
be avoided. The existence of a theoretical justification for the
emergence of the lognormal model as the distribution of claim amounts
is another strong point in favour of that model. On the question of
convenience in use, we believe that, with the wide availability of
computers,, either model is equally suitable.
5.8 Tables
-200
Table (5.1)
Two-parameter inverse Gaussian model fitted to
the accidental damage data by
the multinomial maximun likelihood method
Period of Accident u R% X2 (D. F. )
73 4th Quarter 150.60 96.77 0.6 96.3 (25)
74 1st it 150.07 96.19 0.7 84.1 (25)
74 2nd " 151.82 103.06 0.5 90.2 (25)
74 3rd " 166.33 111.48 0.3 123.5 . (25)
74 4th " 174.40 111.03 1.0 116.4 (26)
75 1st " 173.62 114.91 1.1 113.5 (26).
75 2nd " 180.27 113.72 0.5 . 96.1. .. (26)
R The ratio of the total expected loss statistic, T, to the
total actual cost of claims
201
" Table tß. 2,1)
**- 3-PARAMETER INV. GAUSSIAN DIS, ***
73/4TH QUARTER DATA
E5TIMATi0 4' BY MULTINOMI AL MAX. LIKELI HOOD METHOD :- "C. 18.05 MEW= 1 68.034 LAM DA- 166.536
MEAN= 149.984 S. D .- 168.789
AMOUNT C ACTUAL EXPECTED ' ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A-E)**2/E
1- 30 478, 474. 4. 62. 0.034 31- 60 518. 554. -36. -1638. 2.339 61- 90 461. 437. 24. 1812. 1.318
"91- 120 359. 328. 31. 3270. 2.930 121- 150 239. 247. -8. -1084. 0.259 151- 180 213. 18qß 24. 3972. 3.048 181- 210 148. . 147. 1. 195. 0.007 211- 240 102.. 116. -14. -3157. 1.690 241- 270 91. 92. -11. -2810. 1.315 271- 300 58, 75. -17. -4853. 3.853 301- 330 66. 61.
. 5. 1577. 0.410
331- 360 45. 50. -5. -1727. 0.500 361- 390 39. 41. -2. ' -751. 0.098 391- 420 35. 34. 1. 406. 0.029 421- 450 '34. 29. 5. 2177. 0.862 451- 480 20. 24. -4. -1862. 0.667' 481- 510 29. 20. 9. 4459. 4.050 511- 540 14. 17. -3. -1576. 0.529 541- 570 8.. 15. -11. -3888. 3.267 571- 600 ' 9. 13. -4. -2342. 1.231` 601- 700 29. 30. -1. '-651. 0.033 701- 800 18. 19. -1. -751. 0.053 801- quo 20. 12. 8. 6804. 5.333 901-1000 6. 7. -1. -951. 0.143
1001-1100 4. 5. . -1. -1050. 0.200 1101-1200 4. 3. 11 1150. 1201-1300 1. 2.. -1, -1250. 1301-1400 3. 1. 2. 2701. 1401-1500 1. 1. 0. - 0. 1501-1600 0. 1. -1. -1550. 1601-1700 1. 0. 1. 1650.
1 1701-1800 0. 0. 0. 0, 1801-1900 0. 0. 0. " 01 1901-2000 1. 0. 1. 1950. 2001-2100 0. 0.
. 0. 0.
2101-2209 0. 0. 0. . 0. 2201-2300 0. 0. 0. 0, 2301-2400 ----------
1. ' -----------
0. -------- w..
1. -. r.. r. ýwr-------
2350. -----------
2.000 -------
TnTAL ----------
3045. .
----------- 3044.
---------- -------------- 2645.
----------- 35.397 -------
D. F. = 22 TOTAL EXP. L065
--------------- 0.6 % ii 0.025 TOTAL ACT. COST
202
i
Table (5.2.2)
3-PARAMETER INV. GAUSSIAN DIS. "***
74/1S11' WUARTER DATA
ESTIMATION BY MULTINOMIAL MAX. LIKELIHOOD METHOD : - C= 15.08 MEW= 164.632 LAMDA- 153.065
MEAN= 149.553 0 6.0. = 170.739
AMOUNT £ ACTUAL. EXPECTED ACTUAL- EXPECTED CL, NO. CL, NO. EXPECTED LOSS (A-E)* *2/E
1-" 30 381, 377. 4, 62, 0,042 31- . 60 428, 458. -30. --1365. 1.965 61- 90 35 4. 354. -3. -227. 0,025 91- 120 334. 262, 72, 7596. 19.786
121- 150 211. 196, 15. 2032. 1.148 151- 180 133. 149. -16, -2648. 1.718 1"81- 210 98, 115. -17. -3323, 2.513 211- 240 82. 91, . -9, -2029. 00890 241- 270 54, 720 -18, -4599. 4.500 271- 300 . 52, 58. "-6. . -1713. 0.621 301- 330 53. 48, 5. 1577. 0.521 331- 360 . 36. 39. -3. -1036. 0.231 361- 390 29, 32. -3. -1126. 0,281 391- 420 26, 27, -1. -406, 0.037 421- 450 22. 23. -1. -436. 0.043 451- 480 22. 19. 3,. 1396. 0,474 481- 510 17, 16. 1. 496. 0.063 511"- 540 10, 14, -4" -2102. 1,143 541- 570 19. 12, 7.
. 3888. 4.083
571- 600 4. 10, -6. -3513. 3.600 601- 700 26, 24. 2, 1301. 0.167 701- 800 21, 15, 6. 4503. 2.400 801- 900 11,1 10. 1,, 851. 0.100 901-1000 10. 6. 4, 3802. 2.. 667
1001-1100 5. 4. 1. 1050. 1101-1200 2.. 3. -1. -1150, 0.000 1201-1300 11 . 2. -1. -1250. 1301-1400 2. 1. 1. 1350. 1401-1500 ' 0. 11 -16 . -1450. 1501-1600 0. 1" -10 -1550. 1601-1700 0. U, 06 . 0. 1701-1800 1, 0. 1. 1751. 0.200
------. - ------ .. ------------ ------ "---- .. --. ----------------------
TOTAL 24'41. 2439. 1731. ' 49.2.18 ---------------------"------- r--------------------- -------- -----
TOTAL EXP, LOSS - I' < t1,001 --------------- 0.5 % TOTAL ACT. COST
203
Table (5.2.3)
3-PARAMETER INV. "GAUSSIAN DIS. **
74/2ND QUARTER DATA
ESTIMATION BY MULTINOMIAL MAX. LIKELIHOOD METHOD : - C= 19.46 MEW 170.666 LAMDA= 1R2039
MEAN= . 151.20A S. D. = 164. SH7
0 AMOUNT F ACTUAL EXPECTED ACTUAL- EXPECTED
CL. NO., CL. NO. EXPECTED LOSS (A-E)**2/E 1- 30 351. 316. 5. 78. 0.072
31- 60 3A0. 424. -44. -2002. 4.566 61- 90 382. 345. 37. 2793. 3.968 91- 120 295. 263. 32. 3376. 3.894
121- 150 211. 200. 11. 1490. 0.605 151- 180 142. 153. -11. -1820. 0.791 181- 210 114. 119. -5. -978. 0.210 211- 240 101. 94. 7. 1578. 0.521 241- 270 57. 75. -18. -4599. 4.320 271- 300 51. 6n. -9. -2569. 1.350 301- 330 39. 49. -10. -3155. 2.041 331- 360 36. 40. -4. -1352. 0.400 361- 390 25. 33. -Fl. -3004. 1.939 391-" 420 24. ' 27.
. -3. -1216. 0.333 421- 450 27. 23. 4. 1742. 0.696 4'51- 480 18. 19. -10 -466. 0.053 4R1- 510 21. M. 5. 2477. 1.562 511- 540
"17. 14. 3. 1576. 0.643
541- 570 12. 11. 1. 555. 0.091 571- 600 11. 10. 1. 5F15. 0.100 601- 700 30. 23. 7. 4553. 2.430 701- 800 13. 14.
" -1. -751. 0.071 Sol- 900. 11. 9. 2. 170 1. 0.444 901-1000 4. 5. " -1. -951. 0.200"
1001-1100 7. .. 3. 4. 4202. 1101-, 1200 0. 2. -2. -2301. 1201-1300 1. 1. 0. 0. 1301-1400 1. 1. 0. " 01. 1401-1500 0. 1. -1. -1450. 1501-1600 1. 0. 1. 1550.
"1601-1700 0. 0. o. 0. 1701-1800 0. " 0. 0. 0. 1801-1900 ---- ----
1. ------------
0. ----------
1. -----------
1850. ----- -
1.125
TOTAL
--------- 2383.
------------ 2380.
---------- -----------
- - .
3466. -------
------------- 32.127
- -- "
------------ D. F. 21
TOTAL EXP. LOSS ----------------
y 0. (15 TOTAL ACT. COST
204
Týb1c (5. ". 4)
i *3 -PARAMETER ENV. GAUSSIAN DIS.
74/3RD QUARTER DATA
ES TIMATION BY MULTINOMIAL MAX. LIKELIHOOD METHOD ; - C= 20.92 MEW- 186.695 LAUUDA-, 199.955
MEAN= 165 . 775 S. D. = 180.398
AMOUNT £ ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. 'N0. EXPECTED LOSS (A-E)**2/E
1- 30 362. 356. 6. 93. 0.101 31- 60 427. 462. -35. -1592. 2.652 61- 90 383. 393. -. 10. -755. " 0.254 91- 120 356. 308. 48. 5064. 7.481
. 121- 150 283. 239. 44. 5962. 8.100 151- 180 191. 187. '7. 115B. 0.262 181- 210 137. 147. -10. -1955. 0.660 211- 240 97. 118.1 -21. -4735. 3.737. 241- 270 66. 95. -9. -2299. 0.853 271- 300 71. 77. -6. -1713. 0.468 301- 330 64. 64. 0. 0. 0.000 331- 360 45. 53. -8. . -2764. 1.208 361- 390 44. - 44. U. 0. 0.000, 391- 420 25. 37. -12. -4806. 3.092 421- 450 26. 31. -5. -2177. 0.006 451- 480 22. 26. -4. -1862. 0.615 481- 510 25. 22., 3, . 1486. 0.409 511- 540 14. 19. -5. -2627. 1.316 541- 570 14. 16. -2. -1111. 0.250 571- 600 17. M. 3. 1756. 0.643
" 601- 700 32. 34. -2. -1301. 0.118" 701- 800 34. 21. 13. 9756. 8.048 801- 900 17. 13. 4. 3402. 1.231 901-1000 4. 9. -5. -4752. 2.778
1001-1 100 9. 6. 3. 31 51 . 1.500 1101-1200 4. 4. 0. 0. 0.000 1201-1300 0. 2. -2. -2501. 1301-1400 1. 2. -1. -1350. 140 1-1500 3. 1. 2. 2901. 1501-1600 0. 1, -1. -1550. 1601-1700 1. 1. 0. 0. 1701-1800 0. 0. 0. 0. 1801-1900 0. 0. 0. 0. 1901-2000 0. 0. 0. 00 2001-2100 0.. o. 0. 0. 2101-2200 1. 0. " 1. 2150.
11 2201-2300 "0. U. 0. 0. 2301-2400 0.. 0. 0. 2401-2500 0. 0. 0, 0. 2501-2600 -----------
1. -----------
. 0. ----------
1. " -----
2550. 0. roo
TnTAL
-----------
2799,
----------- ' 2802.
----------
-----
-----------
--------------------- -481.47.401
----------- "
----- D. f=.
------- a 23
TOTAL EXP. LOSS -----.. . -------- - -0 r1
Ci [, !ý f1/'ý/ L: 0.002
TOTAL ACT. nST
205
Table
*** 3-PARAMETER INV. GAUSSIAN DIS. *
74/4TH QUARTER DATA I
ESTIMATION BY MULTINOMIAL MAX. LIKELIHOOD METHOD t- c= 23.99 MEW= 197.730 LAMDA= 211.827
MEAN= 173,740 S . D. = 191.03A
AMOUNT r'. ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A-E)**2/E
1- 30 394.. 390. 4. 62. 0.041 31- 60. 452. 479. -? 7. -1228. 1.522 61- 90 426. 413. 13.
. 982. 0.409
91- 120 348. 329. 19. 2004. . 1.097 121- 150 272. 259. 13. 1761. 0.653 151- 180 219. 205. 1A. 2317. 0.956 181- 210 - 154. 164. -11). -1955. 0.610 21 1- 240 ' . 124. 132. -8. -1804. 0.485
" 241- 270 -105. 107. -2. -511.. 0.037 271- 300 78.. 88. -1n. -2855. 1.136 31)1- 330 75. 73. 2. 631. 0.055 331- 360 58. 61. -3. -1036. 0.14P 361--390 55. 51. 4. 15112. 0.314 391- 420 29. 43. -14. -5677. 4.558 421- '450 43. 30. 7. 3048. 1.361 451- 480 24.. 31. -7. -3258. 1.581 481- 510 22. 26. -4. -19A2. 0.615 5,11- 540 24. 23. 1. 525., 0.043 541- 570 19. 19.. 0.
_ 0. 0.000
571- 600 14. 17. -3. -1756. 0.529 601- 7110 42. - 41. 1. 651. 0.024 701- 800 28. 26. 2. 1501. 0.154 601- 900 20. 17. 3. 2551. 0.529 901-1000 17. 11. 6. 5703. 3.273
1001-1100 8. ,
7. 1. 1050. 0.143 1101-1200 5. 5. 0. a, 0.000 1201-1300 1. 3.
. -2. º -250 1. 1301-1400 5. 2. 3. 4051. 0.200 1401- 1500 0. 2. -2. -2901. 1501-1600 1. 1. 0. 0. 1601-1700 0. 1. -1. -1650. 1701-1800 0. 1. -1. -1751. 1P01-1900 0. 0. 0. 0. 1901-2000 n. 0. 0. 0. 2001-2100 1. 0. 1. 2050. 2101-2200 .
0. n. 0. n 2201-2300 0. 0. 0. () . 2301-2400 1. 0. 1. 2350. 0.800 ---------- ------------ ---------- -------------------------------
TOTAL 30611. 3063. 1875. 21.274
TOTAL EXP. LOSS
---------------- 0.4 y 7 0.10 TOTAL ACT. COST
206
75%1$T QUARTER DATA Table (ßi2.6)
ESTIMATinN BY MULTINt1MIAL MAX. LIKELIHOOD METHOD ; - C= 26.28 MEW= 199.227 LAMM 230.747
MEAN= 172,943 S*Do-. 1A5.120
AMCUNNT. £ ACTUAL. EXPECTED ACTUAL- 'EXPECTED CL. NI O, CL. NO, EXPECTED LOSS (A-E 21G 1- 30 3249
. 322� 2; 31. 0.3 31- 60 387, -399. -12, -546. 1 0.361 61- 90 345. 351. -6. - -453. 0.103 91- 120 289. 284. 5. 528, 0.080
121- 150 253. 225, 28. 37949 3.484 151-. 180 187, 179, 86 1324. 0.358 181- 210
. 1389 143.. -5.,, -978.; 0.175
'211- 240 114. 115. -1, -225. 01009 241- 270 939 949 -1, ' -256. 0,011 271- 300 67. 77. -101 -2855. 1.299 301- 330 63. 63. 0. 00 0.000 331- 360 449 52. -8, -2764. 1.231 361- 390 44, 44,. 0, () . 0.000 391- 420 . 35. 37. -2, .. -811.. 0,108 421- 450 25. 31. -6. -2613. 1.161 451- 480 26. 269 0, 0.000
'481--; 510 18. 22, -4. -1982. 0,727 51.1- 540 18. 19. . -1, -525. 0,053 541- 570.. 22. 16, 6. 3'333. 2.2 50 571- 600 17. 14, 3. 1756, 0.643 601- 700 39. 34. 5, 3252. 0.735 701- 800 19. 21, /-2, -1501. 0.190 801- 900 18. " 13, - 5. 4252, 1,923 901-1000 12. 9. 3. 2851. 1.000
1001-1100 3. 6. -3. -3151. 11500 1101-1200 1.
" 4. -3, --3451'. 1201-1300 1. 2. -1, -1250. 2.667 1301-1400 2. -2. -2701, 1401-1500 . 1. 1, 0, 01
1501-1600 0. -1550.
" 1601-1700 01 00 1701-1800 0, " 0, 0. . 0 1801-1900 00 01 0. 01
'1901-2000 0. Ö. 01 0. 2001-2100 1, 0. 1. 2050. 2101-2200 1, 0, 11 2150, 2201-2300 011 0. 0. () 2301-2400 0. 0. 0, ,
01 2401-2500 04 0. 0, 0. 2501-2600 011 0. 01 0, ' . 26(11-2700 00 0. 0. 01 2701-2800 00 0, 0a 0, 28k') 1-2901) 01 0. 0. 0 2901.3000 08 '0, 0. 0. 3001-3100, 01 0. 0., 01 3101-3200 0. 0. 3201-3300 00 00 3301-3400 0. (1, u. U; 3401-3500 0. 01 00 0: 3501-3600
----------- 1,
----------- 011
------------ 1
---------- 355(), ,
- '0. cloo*
TO1AL -----------
2607. ---- ------ -
2607. ----------- -------~---
----------- 1260, '
----------
---------- 9ei f111) 8P
------mow--
TOTAL EXP. LOSS p>0.440
Treat. r, r, 7. (; ýýT
f Table (5.2.7)
3-PARAMETER- INV. GAUSSIAN CIS, ***
75/2ND QUARTER DATA
ESTIMATION BY MULTINOMIAL MAX. LIKELIHOOD METHOD ; - S 21.49 MEW- 201.211 LAMDA= 203.077
MEAN= 179.717 5. D. = 2009285
AMOUNT E ACTUAL. EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A-E)**2/E
1- 30 302. 299. 3. 46. 0.030 31- 60 374. 390. -16. -728, 0.656 61- 90 332. 336, E -4, -302. 0.048 91- 120 277. 267. 10. 1055. 0.375
121- 150 235. 210. - 25. 3387. 2.97e 151- 180 187. 166-. 21. 3475. 2.657 181- 210 122. 133. -11, -2150. " 0.910 211- 240 110. 108, 2. 451, 0.037 241- 270 80. 88. -8. -2044. 0.727 271- 300. 72. ,
72. 0. 0. 0.000 301- 330 47. 60. -13. -4101. 2,017 331- 360 39. 50, -11. _ -3800. 2.420 361-. 390 40. 42. -2. -751. 0.095 391- 420 38,, 36. 2. 811. 01111 421- 450 29, 30. -1, -436., 0,033 451- 480 21. 26.. -5. -2327, 0,9612 481- 510 30,
. 22. 8. 3964. 2.909
511- 540 19. 19. 0. 0. 0.000 541- 570 $7.. 17, 0. 0. 01000 571- 600 11. 14. --3. -1756. 0.643 " 601- 700 36. 36. 0. 0. 0.00'0 701- 800 22. 23. -1, -751. 0.043 801- 900 22. 15. 7: 5953. 3.267 901-1000 11. 10. 1. 951. 0.100
1001-1100 4. 7. -3. -3151. 1.286 1101-1200 3. 5. -2. -2301. 0.600 1201-1300 3. 3. 0, 0o , 1301-1400 6.
. 2. 4, 5402. 3.200
1401-1500 2,. 2. 0, 0, 1501-1600 1. " 1. 0. 00 1601-1700 0. 1. -11 -1650, 1701-1800' 1. 1. 0, 0l 1801-1900 00 0. 0. 0. 1901-20010 1. 0: 1, 1950. 2001-2100 0. 0. 0. 0. 2101-2200 " 0. "'0. 0,
. U.
2201-2300 -----------
1. ----------
0. - -----
1. ----
2250. 0 . Poo
TOTAL -----------
2495. ----------
2491. ' ------------ -----------
3447. ----------
'--23*101
---------- U. F. m 24
TOTAL EXP. LOSS
. ------ -------- = 0.8 °5 P>0.10 TOTAL ACT, COST
208
Table (5.3.1
*** 3-PARAMETER INV. GAUSSIAN DIG. ***
PREDICTION OF 74/4TH QUARTER CLAIMS Cn ,T USING 73/4TH QUARTER MULT. HAXLIK. PARAMETERS
C= 18.1 MEW= 168.034 LAt1DA= 166.536 INFLATION RATE I=18.2°: CALCULATED FRnt! GENERAL INDEX OF RETAIL PRICES
PREDICTION PARAMETERS ARE :v C=21.4 MEW=19f. 616 LAMDA=156.846 MEAI1 CLAIM AMOUNT= 177.22 S. D. =199.51
ACTUAL 74/4TH PARAtETERS :- C=24.0 MEY! =197.730 LAMDAm211.827 UEAN CLAIM AMOUNT= 173.73 S. D. -191.04
AMOUNT £ ACT. NO. EXP. NO. A-E EXP. LOSS (A_E)**2/E 1- 30 394. 386. A. 124. 0.166
31- 60 452. 486. -34. -1547. 2.379 61- 90 426. 413. 13. 982. 0.409 91- 120 348. 326. 22. 2321. 1.485
121- 150 272. 255. 17. 2304. 1.133 151- 180 219. 202. 17. 2813. 1.431 181- 210 154. 161. -7. -1368. 0.304 211- 240 124. 130. -6. -1353. 0.277 241- 270 105. 106. -1. -256. 0.009 271- 300 78. 87. -9. -2569. 0.931 301- 330 75. 72. 3. 947. 0.125 331- 360 58. 60. -2. -691. 0.067 361- 390 55. 51. 4. 1502. 0.314 391- 420 29. 43. -14. -5677. 4.558 421- 450 43. 37. 6. 2613. 0.973 451- 480 24. 31. -7. -3258. 1.581
, 481- 510 22. 27. -5. -2477. 0.926 511- 540 24. 23. 1. 525. 0.043 541- 570 19. 20. -1. -555. 0.050 571- 600 14. 17. -3. -1756. 0.529 601- 700 42. 43. -1. -651. 0.023 701--800 28. 28. 0. 0. 0.000 1101- 900 20, 18. 2. 1701. 0.222 901-1000 17. 12. 5. 475?. 2.0113
1001-1100 A. 8. 0. 0. 0.000 . 1101-1200 5. 6. -1. -1150. 0.167
1201-1300 1. 4. -3. -3751. 1301-1400 5. 3. 2. 2701. 1401-1500 0. 2. -2. -2901. 1501-1600 1. 1. 0. 0. 1601-1700 0. 1. -1. -1650. 1701-1800 0. 1. -1. -1751. 1801-1900 0. 1. -10 -1850. 1901-2000 0. 0. 0. 0. 2001-2100 1. 0. 11 2050. 2101-2200 0. 0, 0, 0. 2201-2300 0. 0. 0. 0. 2301-2400 1. 0.. 1. 2350.
------------ TOTAL ---------
3064 ----------
3061 ------ ---------- -7528.
---------- 20.185
------------------------------------- --------------------
CHI SQ. STAT. - 21.825 . D. E. = 27 F>0 10 , . TOTAL ACTUAL COST - 533707. TOTAL EXPECTED COST s 541234". i'OL - SI: T(NOV I= 0.015
1> 0.20 ý TOTAL EXP, LOSS ---------------- - 1.41 % TOTAL ACT, C05T 209
*** 3-PARAMETER INV. GAUSSIAN DIS. # "*
PREDICTION RF 75/13T QUARTER CLAIMS COST Table (5.3.2)
USING 74/13T t JARTER MULT. MAXLIK. PARAMETERS
C= 15.1 MEW- 164.632 LAMDA= 153-065 INFLATION RATE I=20.3°', CALCULATED - FROM
. GENERAL INbEX GF RETAIL PRICES
PREDICTION PARAMETERS ARE :- C-1A. 2 MEW=198.052 LAMDA=: 184.137 MEAN CLAIM AMOUNT= 17909 S. D. =205.40 ACTUAL 75/1ST PARAMETERS
.- C=27.6 MEW=199.260 LAMDA=239.515 MEAN CLAIM AMOUNT= 171.66 S. D. =181.75
AMOUNT C ACT. NO. EXP. NO. A-E EXP. LOSS (A-E)**2/E 1- 30 324. 314. 10. 155. 0.318
31- 60, 387. 423. -36. -1638. 3.064 61- 90 345. 356. -11. -E831. 0.340 91- 120 289. 278. 11. 1160. 0.435
121- 150 253. 216. 37. 5013. 6.3311 151- 180 187. 170. 17. 2813. 1.700 181- 210 138. 136. 2. 391. 0.029 211- 240 114. 109. 5. 1127. 0.229 241- 270 93. 89. 4. 1022. 0.160 271- 300 67. 74. -7. -1990. 0.662 301- 330 63. 61. 2. 631. 0.066 331- 360 44. 51. -7. -2418. 0.961 361- 390 44. 43. 1. 376. 0.023 391- 420 35. 37. -2. -811. 0.108 42_1- 450 25. 31. -6. -2613. 1.161 451- 480 26. 27. -1. -466. 0.037 481- 510 18. 23. -5. -2477. 1,087 511- 540 18. 20. -2. -1051. 0.200 541- 570 22. 17. 5. 2777. 1.471 571- 600 17. 15. 2. 1171. 0.267 601- 700 39. 38. 1, 651. 0.026 701- 800 19. 25. -6. -4503. 1.440 801- 900 18. 16. 2. 1701. 0.250 901-1000 12. 11. 1. 951. 0.091
1001-1100 3. 8. -5, -5252. 3,125 1101-1200 1. 5. -4. -4602. 3.200 1201-1300 1. 4. -3. -3751. 1301-1400 0. 3. -3. -4051. 1401-1500 1. 2. -1. -1450. 1501-1600 0. 1. -14b -1550. 1601-1700 1. 11 0, 0. 1701-1800 0. 1. -1. -1751. 1801-1900 0. 1. -1. -1850. 1901-2000 0. 0. 0. 0. 2001-2100 11 0. 1. 2050. 2101-2200 11 0. 1. 2150, 2201-2300 0. 0. 01 0. 2301-2400 0. 0. 0. 0. 2401-2500 0. 0. 0. 04 2501-2600 0. 0. 0. 0. 2601-2700 0. 0. 0. 0. 2701-2800 0. 0, 0. 0. 2801-2900 0. 01 0. 0. 2901-3000 0. 0. 0. 0. 3001-3100 0. 0. 0. 0. 3101-3200 0. 0.. 0. 0, 3201-3300 0. 0. 0. 01 3301-3400 0. 0. 0. 0. 3401-3500 0. 0. 0. 0. 3501-3600
-------------
1. --------
0. ----------
1.
------ 3550.
----------- --------- TOTAL 26t)7 2606 -15375.26.809
Table (5.3.2) - continued
CHI SQ. STAT. °32.619 , D. F. =27 9 P70. ß0
TOTAL ACTUAL COST = 452n59. TnTAL EXPECTED COST = 467433. KOL - SM. IRROV D=0.15
TOTAL-EXP. LOSS P> 0.20
---------------= -3.40 % TOTAL ACT. COST,
211
11 Table (5.3.3) 3-PAfAMETER INV. GAUSS-AN DIS.
PREDICTION OF 75/2ND QUARTER CLAIMS CnST USING 74/2ND QUARTER MULT. IMAXLIK. PARAMETERS
C= 19.5 MEW= 170.666 LAMDA= 182.839 INFLATION RATE I=24.3° CALCULATED FROM GENERAL INDEX OF RETAIL PRICES
PREDICTION PARAMETERS ARE :- C=24.2 UE'W=212.138 LAMDA=227.269 MEAN CLAIM AMOUNT= 187.90 S. D. -204.95
ACTUAL 75/2ND PARAMETERS :- C=21.5 MEW=201.211 LAMOAa203.077 MEAN CLAIM AMOUNT= 179.71 S. D. =200.28
AMOUNT F, ACT. NO. EXP. NO. A-E EXP. LOSS (A-E)**2/E 1- 30 302. 273. 29. 450. 3.081
31- 60 374. 366. 8. 364. 0.175 61- 90 332. 326. 4. 302. 0.049 91- 120 277. 267. 10. 1055. 0.375
121- 150 235. 214. 21. 2845. 2.061 151- 180 187. 171. 16. 2648. 1.497 181- 210 122. 138. -16. -3128. 1.855 211- 240 110. 113. -3. -677. 0.080 241- 270 80. 93. -13. -3321. 1.817 271- 300 72. 77. -5. -1427. 0.325 301- 330 47. 64. -17. -5363. 4.516 331- 360 39. 54. -15. -5182. 4.167 361- 390 40. 45. -5. -1877. 0.556 391- 420 38. 38. 0. 0. 0.000 421- 450 29. 33. -4. -1742. 0.485 451- 480 21. 28. -7. -3258. 1.750 481- 510 30. 24. 6. 2973. 1.500 511- 540 19. 21. -2. -1051. 0.190 541- 570 17. 18. -1. -555. 0.056 571- 600 11. 16. -5. "-2927. 1.562 601- 700 36. 39. -3. -1951. 0,231 701- 800 22. 25. -3. -2251. 0.360 801- 900 22. 16. 6. 51913. 2.250 901-1000 11. 11. 0. 0. 0.000
1001-1100 4. 7. -3. -3151. 1.286 1101-1200 3. 5. -2. -2301. 0,800 1201-1300 3. 4, -1. -1250. 1301-1400 6. 2. 4, 5402. 1401-1500 2. 2. 0. 0. 1501-1600 1. 1. 0. n. 1601-1700 0. 1. -1. -1650. 1701-1800 1. 1, 0. 01 11301-1900 0. 0. 0. 0» 1901-2000 1. 0. 1. 1950, 2001-2100 01 0. 0. 0. 2101-2200 0. 0. 01 n, 2201-2300
------ 1.
---------- 0.
---------- 1.
------ 2250.
--- ----- TOTAL
----------- 2495
--------- 2495
----------- ------
------- -17725.
----------
----------- 31.021
-----------
CHI SO. STAT. " 352.72 ,0F. ¢ 27 , r> 0. to TOTAL ACTUAL CnIT = 4493118. TOTAL EXPECTED COST - 467033. KOL - =r'ýý-ý'cv ýý 0.035
TnTAL EXP. LO; CS P <O. O1
---------------r -3 -94 % TOTAL ACT. CO 9T 212
CHAPTER 6
The Pareto Distribution
6.1 Introduction
The Iareto distribution, which is positively skewed and has a long
tail, was originally proposed by Vilfredo Pareto as a model for the
distribution of incomes over a population. It has also been successfully
used to model empirical distributions of claim amounts in general
insurance and, in particular, in fire insurance. Benckert and
Sternberg (1957), Anderson (1971) and Benckertand Jung (1974) have
used this distribution as a model for fire insurance claim amounts.
In this chapter we will study this model and will consider its
application to the accidental damage claim amounts. The Pareto
distributions of the first and second kind will be initially defined
and some of their properties will be studied. A graphical test will be
discussed for the purpose of determining whether a given sample is likely
to be from a Pareto population. We will then apply this test to our
accidental damage data and will, in the event, show that the Pareto
distribution is not a satisfactory rmdel. The estimation problem is
next dealt with and the effects of inflation on the parameters of the
model will be discussed.
6.2 Definition
A random variable X is said to have a Pareto distribution of the
first kind if its distribution function, denoted by P1(x; A, B), is of
the form A
P1(x; A.. B) =1- (i. ) A, B > O, x. B
This is a special form oE Pearson Type Vl distribution. The probability
density function (p. d. f. ) of X is
fi`A+1 fý, ý (x; A, B) =Bt Y) A, 13 > 0, B (G . 2-21
213
where A and B are the sham and scale parameters respectively.
A random variable X is said to have a Pare-to distribution of the second
kind if its distribution function, denoted by P2(x; C, A,. B), is of he
. bn
P2(x; C, A, B) =1- ()A A, B > 0, x >. B-C (6.2-3)
This is sometimes called the Lomax distribution and is also a Pearson
Type Vi. it differs fron Pl(x; A, ß) by having the location parameter
C which acts as a threshold below which the values of X are not
realized. The p. d. £. of X is given by
A+1 fß, 2 (x; C, A, B) =C iB -ff A, B > 0, x3B-C (6.2-4)
6.3 Properties of the Pareto Distribution
In this section we will mention some of the properties of the
Pareto distribution of the first kind. For more details reference should
be made to, for instance, Johnson and Kotz (1970).
For the distribution P1(x; A, B) it can be shown that the rth moment of
X about zero is
r E(XT) _
Tr (if A> r) (6.3-1)
Hence the mean of the distribution of X is
E(X) = AB (if A> 1) A-1 (6.3-2)
The variance of X can be shown to be 2
var(X) _ AB (if A, 2) (6.3-3)
(q-i) (A-2)
Therefore the mean and variance of tho distribution exist when A>1
and A>2 respectively.
The coefficient of variation of X is x where
214
1
2A=1 (if A- 2) A(A-2)
(6,3.4)
which is a function of A, the shape parameter, only. As the mean of
the distribution does not always exist, the single mode of the distribution
which is located at B may be used in its place.
The median of the distribution is at
Xmedian = B(2)i/A (6.3-5)
In general the quantile of order q is xq : where
X= B(1 - q) -'/A
O<a<1 (6.3-6) a
The properties of the Pareto distribution of the second kind can
be obtained from those of the first kind since the former distribution
is derived from the latter by the transformation x -" x+C. This is
a translation which leaves the shape of the frequency curve of P1(x; A , B)
unchanged and only shifts it by an amount -C along the x-axis. The
measures of central tendency are, therefore, decreased by C while
the moments about the mean and the measures of dispersion remain the
same as for the first kind.
6.4 The Graphical Test for the Pareto Distribution
It is possible to test graphically if a given sample is likely
to be from a Pareto population. For the distribution of the first
kind given by (6.2-1) we can show that
log(1-P1) = -A log x+A 1og B (6.4-1)
where P1 = P1 (x; A, B)
Therefore the locus of the points (logx , log(l - 11)) is a straight
line whose gradient and intercept are A and Alogl3 respectively. LCt
us assui e that we are giver. a samplo of n independent rundem
215
observations from n Pareto distribution. If we define the sample
empirical distribution function as
F(x) = the proportion of observations 6x (6.4-2)
then we can use F(x) as an estimate of Pl(x; A, B) at point x. Hence
if we plot the points (log x, log(1 - F(x))) on a rectangular system
of co-ordinate axes, or plot(x, l-F(x)) on logarithmic axes, we should
find that the points lie approximately on a straight line. If the
points do not appear to lie approximately on a straight line, we conclude
that the sample is not from a Pareto population of the first kind.
For the Pareto distribution of the second kind, equation (6.4-1)
becomes
log (1 - P2) = -A log (x + C) +A log B (6.4-3)
where P2 = P2(x; C, A, B)
Hence the locus of the points (109(x + Q, log (1 - P2)) is a straight
line. When C is unknown the first step is to assinne C=0 and to plot
the resulting points (as in the case of the 3-parameter lognormal and
Weibull distributions). If the sample is from a population with C not
equal to zero, judging by the curvature of the resulting curve, we
should choose another value for C such that the points are rectified
to lie approximately on a straight line. If such aC exists, we conclude
that the sample is from a Pareto population of the second kind and
riay use this value of C as a graphical estimate of the unknown parameter
C. This technique was described, in detail, in sections 3.11 and 4.4
for the 3-parameter lognormal and Weibull distributions respectively.
6.5 The Pareto Graphical Test on AD Dzta
To test if the Pareto distribution can be used as a model for the
accidental damage (AD) data we wrote coiputer program P22. From a
216
given saiiple of data, the program calculates the empirical distribution
function F(x) at each sample value x and then plots the points
(log x, log (l -F (x)) . For the seven samples of Al) data which were
given in tables (1.1) to (1.7) the graphs are presented in figures
(6.1-a) and (6.1-b). It is observed that for each sample the points
lie approximately on a curve which is far from a straight line. The
points in the upper tail of the sample values lie almost on a straight
line, but the curvature in the lower tail values is so marked that even
an addition of C to the values of x( and hence a Pareto distribution
of the second kind) does not seem likely to rectify the points enough
to lie on a straight line. We, therefore, conclude that the Pareto
distribution is not a possible model for the distribution of our AD
claim amounts. However, as indicated by the graphs, such a model may
very well fit the distribution of the larger claims in the sample.
217
9
8
Str*oIght (3' plot oG PARETO DIS. For 7S/2ND QUARTER DATA
ICI -ttrcl. "r.. »
1j Ulraly'tl ltn+" plot oý 1'AI: CTC UI;.
7$'IST OUAFIIK MMA
LOUx)
6
3" "
s
f
3
i
2i
' *"
f f
f e
-; __-T"_---. 21C ei LI""f r)
171S67E Lx.. s>
6.6 Estimation of the Parameters of the Pareto Distribution
Although the results of the previous section were not encouraging,
for the sake of completeness of the present work, we deal with the
problem of estimation from grouped data. Several methods of estimation
such as the methods of moments, quantiles, graphical, least squares
and maximum likelihood have been studied by various authors and a
review is provided in Johnson and Kotz (1970). Some of these methods
are directly applicable for estimation from grouped data while others
(like the noments, quantiles and maximum likelihood) can be modified
for this purpose in the same way as we described for the 2-parameter
lognormal distribution. In this section, we explain briefly the method
of least squares and them concentrate on the multinomial maximum
likelihood (. &NL) method.
The least squares method is based on the relationship (6.4-1).
Instead of fitting a straight line to the points (log x, log(1-F(x)))by
eye, we use the least squares technique to find the unknown coefficients
of the line. For Pl(x; A, B) this is straightforward. For P2(x; ' C, A, B)
when C is unknown, in order to avoid solving a system of non-linear
equations in the parameters, we can use the computing technique which we
suggested for the 3-parameter lognormal and Weibull distributions in
sections 3.12.1 and 4.6.1 respectively.
The AML is a very suitable method of parameter estimation from
grouped data and it produces efficient estimators. Estimation of the
parameters of the Pareto distribution of the second kind, P2(x; C, A, B),
will be dealt with here but the same argent applies for Pl(x; A, B)
by putting C=0 in the following exposition.
Let us assune that we have a sample. of grouped data where n independent
random observations on a random variable X (in our case he claim
amount) have. been grouped according to t1heir size into k nni. tually exclusive
218
intervals. Iurther, let nl be the number of observations (claims) in
the class interval (x i-1 xi) for i=1,2, ... ,k, such that
k n= ni
i=i
If X is distributcd as P2(x; C, A, B), given by (6.2-3) , i. e.
P2 (X; C, A, B) =1- (X$` )A (6.6-1)
then let pi be the probability that an observation (claim) occurs
in the interval (xi_1, xi), i. e.
pi = P2(xi; C, A, B) - P2(xi-1; C, A, B) (6.6-2)
for i=1,2, ... ,k
from (6.6-1) it follows that
pi. = x1B + )A - (1
x+C )A (6.6-3)
therefore the sample likelihood function will be proportional to L where 1i yak r.. ". ... ý... ".. .ý
k L= pi
ni (6.6-4) i=1
and the loglikelihood function will be
k log L= nllog p1 (6.6-5)
i=1
from (6.6-3) it follows that
k log L + n1 log [(x±11+c)A
)A (6.6-6)
AA
The naxirzl likelihood cstioates, (C ,A, B), of the parameters are
obtained by maximizing logL sir: ultaneý, usly with respect to A, B and C.
To avoid solving; non-linear equations (which is a laborious task when k,
219
the number of intervals, is large) we suggest the computing tecxMique
described in section 3.7.7 for maximising log L with respect to A, B
and C. To start the iteration process we can use a set of estimates
found by one of the simpler, but less accurate, methods. For instr. nce,
the moments, quantiles or graphical estimates (or their combination)
may be used. The set (C, A, B) which ma irrizes log L will be the
required DINS, estimates.
The graphical tests showed earlier that the Pareto distribution does
not represent our AD data. Therefore, we will not attempt to fit
this model to the complete histogram of the data. However, figures
' (6.1-a) and (6.1-b) showed that the points corresponding to the upper
tail of the distribution of claim amunts lie approximately on a
straight line. In chapter 7 we will fit this model to the upper tail
of the AD histograms and will compare its fit with that of the truncated
lognonnal distribution.
6.7 The Effects of Inflation on the Parameters
Following the argument of chapter 3 abcut the effects of inflation
on the parameters of a model, it is considered important to study
these effects theoretically for the Pareto distribution. We will need
to allow for such effects when predicting the future distribution of
claim amounts for a class of general insurance business in which the
Pareto model has been found to represent the distribution of claim
amounts.
Let us again assume that the effect of inflation is to increase a
claim of amount X to U= X(1, + i) over a period where i is the effective
rate of inflation for that period. If Xis distributed as a Pareto
distribution of the second kind with p. d. f. fP2(x; C, A, B) then by a
simple trarsfoiration of variables we can show that U is distributed as
22.0
a Pareto distribution of the second kindwith p. d. f. fP2(u; rC, A, rB)
where r=1+i. Therefore inflation affects the parameters C and }3
but leaves the shape parameter A unaltered. The results for the
Pareto distribution of the first kind can be deduced from the above by
putting C=0.
6.8 Conclusions
The Pareto distribution was studied in this chapter as a model
for the distribution of claim amounts in general insurance and, in
particular, for the accidental damage claim amounts.
The Pareto distribution, as well as other positively skewed distributicns
like the lognormal, has been shown to represent empirical distributions
of many socio-economic and other naturally occurring quantities. In
section 6.1 reference was made to the uses of Pareto distribution in
fire insurance. A graphical procedure was described which may be used
to test if a sample is likely to be from a Pareto population. Our AD
data proved not to be from such a population. However, points
corresponding to the upper tails of the histograms appeared to lie
approximately on a straight line as shown in figures (6.1-a) and (6.1-b).
This indicates that the tail of the distribution of claim amounts may
well be modelled by the Pareto distribution. As a matter of fact, Johnsen
and Kotz (1970) mention that lognormal distribution fits well the
distribution of income over a large part of the income range but diverges
markedly at the extremities. On the other hand, it has been observed
that the fit of the Pareto curve is rather good at the extremities
of the income range but the fit over the whole range is often rather poor.
If we consider that income and claim amount are to some extent correlated
as represented by the sure at risk, ther.. the above conclusion from the
n aphical tests about the tail of the distribution of our AD claim amounts
becones plausible.
rI
I
z'ý
We proposed the NEC rretl-iod of parameter estimation from grouped data
but did not apply it to the AD data in this chapter. In chapter 7 this
method will be used to fit the Pareto model to the tails of our AD
samples.
The effects of inflation on the parameters were studied and it was
shown that both C and B are increased while A, the shape parameter, remains
unchanged.
222 hh,
CHAPTER 7
Truncated Lognormal Distribution
7.1 Introduction
Often, in practice, data only of claim amounts above a particular
sum are available. For example, the excess of loss reinsurer has
only access to data of claim amounts above the retention point, E.
Such data, for a particular class of business, may be represented by
the incomplete (or truncated) histogram in figure (7.1).
N
c- U
44 0
Figure (7.1) - A"truncated distribution fitted to a histogram of inconiplete data
As another example, the above histogram may represent claim amounts
data, available to an insurance co1Tpany, in respect of a class of
insurance business where the policy holder pays any loss up to the
'voluntary excess' : mount E. The histogram in figure (7.1) can, therefore,
represent the empirical claim arrotmt distribution when it has been singly
223
Ii claim amount
truncated fron below at the 'point of truncation' E.
If we lclow that a particular statistical distribution represents the
complete distribution of claim amounts, then we may be able to fit a
truncated form of that statistical distribution to the incomplete data
as represented in figure (7.1). Essentially, we find a model with
probability density function f whose tail fits the incomplete empirical
distribution (see figure (7.1)). That part of the curve f which lies to
the right of the truncation point E is the frequency curve of the
truncated distribution. Barding (1968) considers the truncated lognormal
model for the distribution of excess of loss reinsurance claim amounts
of motor business.
The studies of the previous chapters of the present work showed that
the lognormal model represents the distribution of our accidental damage
(AD) claim amounts. The 2-parameter lognormal produced a 'better' fit
.n the sense of smaller Chi-square statistics) to the actual data than
the 2-parameter inverse Gaussian distribution. Therefore, in this chapter
we shall consider the truncated lognormal distribution. As we are only
interested in the tail of the distribution, only the 2-parameter model
will be dealt with. Initially, the truncated lognormal distribution will
be defined and some of its properties will be mentioned. Then the problem
of estimation of the parameters from grouped data will be considered
and the multinomial maximum likelihood method will be proposed. By
using this method, two problems' related to our AD data will then be
studied ; these are discussed below :-
1- In chapter 3 we noticed that the significant differences between
the 2-parameter lognoir. ýal distribution and the actual sample values were
caused by large contributions to the Chi-square statistic by one or two
of the intervals in the larger tail values. In that chapter we indicated
that the disagreement could-be due to the inaccuracies and incorpleteness
224
of the data in the lower tail intervals because in order to re, -, lain
entitled to their substantial no-claim discounts some policy holders
do not claim. small amounts. As the claims in the lower tail are
financially less important, it was suggested that the data should be
truncated at the upper boundary of these intervals to see if a more
exact fit to the rest of the data would be obtained. Therefore,, in
this chapter, we will fit the truncated lognormal distribution to samples
of our AD data, as presented in tables (1.1) to (1.7), but truncated
at 130 or £60 or £90. The results-for different points of truncation
will be examined and compared with those of chapter 3 for the complete
samples.
2- The larger claims form the tail of the claim amount distribution
and because of their financial importance a higher degree of agreement
between the model and the empirical distribution is required in this
region. Therefore, it is important to find out if a truncated distribution
fitted to the tail of the histogram of sample values produces a 'better'
fit. For our AD data we truncate the samples at 1600 and fit the
truncated lognormal model to the tail of the actual sample values. The
results will then be compared with the tails of the fitted lognormal
distributions in chapter 3 to see if, in fact, truncation does produce
a 'better' fit in this region.
The histograms of our AD samples in figures (3.5.1) to (3.5.7) indicate
that the Pareto distribution with its mode at 1600 Is likely to fit the
empirical distribution of claim amounts greater than 1600.. The graphical
tests of chapter 6 also implied a Pareto model for the tail of our AD
histograms. The Pareto distribution is much simpler to fit and develop
mathematically than the truncated lognormal distribution. Therefore, to
compare the performance of these two models we will, in this chapter, also
225
fit the Pareto distribution to samples of claim amounts greater than
£600.
As was mentioned previously, the excess of loss reinsurance claim 11
amounts data are available only in the truncated form and hence the treat-
ment of the present problem is applicable in that context.
The findings of this chapter will be summarized in the conclusion
section and the tables will be presented in section 7.6.
'7.2 Definiticn'and'Sdme Properties of'the'Truncated Lognormal Distribution
We say a random variable X has a truncated lognormal distribution 2
if its distribution function, denoted by TLN(x, E; u, o ) is of the form
222 TL1ý1(x, E; µ, ý)
I; N(x; i; d) -L (E; u, Q) for x>E (7.2-1) 1- LINE ;u, a)
=0.......................... otherwise
where E is the point of truncation and LN ("; u, o 2) is the distribution
function of a lognormal random variable, with parameters u and a2 , as
defined in chapter 3 by (3.2-4). Hence the probability density function
(p. d. f. ) of the truncated lognormal distribution, which we denote by
£TIN(X, E ; u, a ), is :
2
2 fLN (X; u"7 ) fTLN(x, E; u, a )= for x>E (7.2-2)
1-LN(E; u, a2)
where fLN ; 11,02 is the p. d. f. of a lognormal random variable, with 2
parameters u, a, as given by (3.2-3).
The rth moment of X about zero can be shown (see for instance,
Aitchison and Brown (1957)) to be
E(Xr) = exp(r u± ýr2a. )Z '- LN(x.; u+ ro?, a21 (7.2-3)
1- LNN(E u>o2)
L26
12 1-1 From (3.3-3) we lo ow that exp(ru +2ra) is the rth moment about zero
of the 2-parameter lognormal distribution. The remaining expression
on the right land side of (7.2-3) can be expressed in terms of the
standard normal distribution function by using relationship (3.2-4).
Hence Quensel (1945) expresses (7.2-3) as :
(rth moment if not truncated) 1- N(U -ra; 0,1) (7.2-4) 1-N(U; 0,1)
where U= log E and N( "; 0,1) is the standard normal
distribution function.
The mean of the distribution of X is, therefore,
E(X) exp (u+ 21 2) iu -Q; 0,1) (7 . 2-s) 1 -N(U ; 0,1)
and the variance of X can be shown to be
22 1 ^exp (u+ )2 Var(X) = exp(Q ) [1-N(U-2a; 0,1)] [1-N(U; 0,1)] - 1- N(U; 0,1)
21
(7.2-6)
7.3 Estimation of the Parameters
In this section we study the problem of estimating the parameters
ý, and c2 for a TL. N(x, E ; u, ß2) dist: ibuticn with the 1aown point of
truncation E. This problem has been dealt with by several authors.
Aitchison and Brown (1957) describe the maxiniun likelihood method when
values of the individual observations in the sarple are available. They
suggest using the tables given by Hald (1949) for carrying out the
inverse interpolation required in this method. Harding (1968) explains
this method for fitting the truncated lognormal modal to the third party
227
excess of loss reinsurance claim amounts of n for insurance business.
The quartile, and graphical methods are not applicable since an 2
estimate of LN(E ; u, Q ) (i. e., the area under the complete frequency 2
curve of 11\1(x ; 1,, a ) and to the left of the truncation point) is
required but there is no information on this.
The method of moments may be used which involves putting the first and
second sample moments equal to their corresponding theoretical values
given by (7.2-3). Ho' ever, there is no easy way of solving the resulting 2
equations for u and a except by trial and error. Grundy (1952) has
considered estimation, from grouped data, by the method of moments for
the truncated normal distribution. His method may be used for the
truncated lognormal distribution since log x is normally distributed. He
gives expressions for adjusting the first and second sample moments about
zero when they are calculated from grouped truncated data.
Tallis and Young (1962) have considered estimation from grouped data by
the nultinomial maximum likelihood, (. »L), method but their technique
involves solving iteratively a system of non. -linear equations in the
parameters.
We will adopt the AIINL method but will apply the computing technique,
described in section 3.7.7, of maximizing the loglikelihood function
iteratively with respect to the parameters.
Let us assume that we have a sample of grouped data where n
independent random observations-(clams) on a random variable X (claim
amount) have been grouped according to their size into k nutually
exclusive intervals. Further, let us assume that each claim is for an
arnount greater then E (the truncation point) and that ni is the number
of claims in the class interval (xi_1, xi) , for i=1,2, ... ,k
and with .: o = r, such that
2 L8
k n= ni
i=1 2
If X is distributed as a TLN(x, E ; ;,, Q ) distribution given by (7.2-1),
then let pi be the probability that the size of a random observation
(claim) will fall in the interval (xi-1, xi)' i. e.
22 pi = TLN(xi, E; u, Q )- TLN(xi-1' E'i, ) (7.3-1)
From (7.2-1) it follows that
LN(xi; u, Q2) - LN(xi-1 ; u, Q2) (7.3-2) pi -
1-LN(E; u, Q2)
Therefore, the sample likelihood function will be proportional to L
where k
_ TT n.
Lýj pi 1 (7.3-3) i=1
and the loglikelihood function will be
k log L ni log pi (7.3-4)
i=l
From (7.3-2) and remembering that from (3.2-4)
2 logx- u LN(x ; u, a )=N(Q; 0,1)
we can write (7.3-4) as :
+u logt =-n log(N ( Log E 0,1)) + 0
k log x. -ux. u + ni1og NC a; o, 1/ - N(log
of ; o'i/
(7.3-5)
2 The maxini^i likelihood estimates (p, o) of the parameters are obtained
by maximizing log L sinultaneously with respect to u and a2 . To avoid
solving non-linear equations in the parameters wo suggest the computing
technique described in section 3.7.7 for maximizing log L with respect to
229 kh.. -
2 u and a. To start the iteration process we can use the estimates of
2 u and a w1-ich we found for the complete samples in chapter 3. If such
a 1nowledge about the possible values of u and a2 is not available they.
the method of moments should be used to work out a set of ( u, a2 ) for
starting the iteration process. In any case, the penalty for choosing
a very inaccurate set of (11, Q2) as starting values is a few more iterations,
which is not really a problem when the computer is used.
7.4 Applications to the AD Data
Computer program P23 was written to estimate the parameters of the
truncated lognormal distribution by multinomial maximum likelihood, (MIL),
method. The complete samples of the AD data which were presented in
tables (1.1) to (1.7) are input to the program and a truncation point is
specified. in chapter 3 we showed that for our data the estimates of the
parameters of the complete 2-parameter lognormal distribution were A 4.5 and v2 1. These values are specified as initial values
of the parameters to start the iteration process.
We first dealt with problem 1 put forward in section 7.1, i. e.,
truncation in the lower tail sample values. Each sample of AD data was
truncated respectively at E= 30, E= 60 and E= 90, and the parameters
of the truncated model were estimated. In each case the computer program 2
gave the estimates of p and a and an extensive table of relevant statistics.
As an example, for the 73/4th quarter data the results of the computer
runs are given in tables (7.1), (7.2) and (7.3). The results for all
the samples have been summarized in tables (7.4) to (7.7). The correspon-
ding results for complete samples, which were given in tables (3.11) to
(3.17), are also reproduced for the sake of comparison. Table (7.4)
gives the estimates of p. We notice that truncation has given rise to
larger values of u compared with the corresponding values for complete
230
samples. Fora 2 we can see, from table (7.5), that truncation has
resulted in smaller values. Tables (7., I) to (7.3) show that the 2
contribution to the Chi-square statistic, X, by different intervals is 2
generally small. The values of X statistic for complete and truncated 2
samples are presented in table (7.6). For complete samples the X
values showed significant differences, at 0.05 level, between the 2
LN(x ; u, a ) model and actual sample values for every period of accident.
The values of these statistics for truncated samples are considerably
smaller and, except for 74/1st and 74/3rd quarters, do not show any
significant differences, at 0.05 level, between the actual sample values
and the truncated fitted models. Table (7.7) gives the ratio of the
total expected loss statistic, T, to the total actual cost of claims.
We observe that smaller values have been obtained for truncated samples
indicating a better overall agreement between the model and actual
sample values.. The above analysis shows that truncation in the lower
tail sample values of the AD data gives rise to a more exact fitting
statistical model compared with the 2-parameter lognormal distribution
for the complete sample. However, the fit does not seem to improve when
truncation is carried on beyond the first interval (E = 30). In terms
of the Chi-square goodness-of-fit test statistics, the results for the
truncated (E = 30) and 3-parameter lognormal (see tables (3.38) to (3.44))
distributions are very similar. The latter distribution is, however,
easier to work with and to develop mathematically. For the AD data, we,
therefore, recommend the use of the 3-parameter lognormal distribution.
The second problem we put forward in section 7.1 was the application
of the truncated lognormal as a model for the distribution of large AD
claim amounts. We, therefore, used progra'n P23 with our samples of AD
data. Each sample was truncated at f; 1600 and (u = 4.5, a' L 1) were
specified as starting values for the iteration process. rar each sample
231
the computer program produced the estimates of u and v2 as well as a
table of relevant statistics. The results are presented in tables
(7.8.1) to (7.8.7). We can see that u is larger than for complete samples
while a2 is considerably smaller. The actual means and standard deviations
of'the truncated samples are given in table (7.11). From tables (7.8.1)
to (7.8.7) we can see that the mean claim amounts of the fitted models
are approximately equal- to their corresponding sample values. The standard
deviations of the models are greater than their actual values but this
could be because we are fitting a curve with an infinite range to a sample
which has a finite range. From the tables of results it is apparent
that the contributions to the Z
statistic are generally small and that
the overall X2 statistics, except for 74/3rd quarter, do not indicate
any significant differences between the model and actual sample values.
The total expected loss statistics are small and are at most 5% of the
total actual cost of claims. This indicates an overall satisfactory fit
of the model to the sample values.
To see how the truncated lognormal model performs against the, mich
simpler to use, Pareto distribution, we wrote computer program P24 to
estimate, from a truncated sample, the parameters of the Pareto distribution
of the first kind. This program uses the MhL method of estimation.
The program was then modified to estimate the parameters of the Pareto
distribution of the second kind. In each case the estimates of the
parameters by the method of moments were input as starting values for
the iteration process. These programs were run with truncated samples
(at B= 600) of our AD data. For a particular sample, 73/4th quarter,
the results are given in tables (7.9) and (7.10). A summary of the results
for all samples are presented in tables (7.12) and (7.13). For the
Pareto distribution of the first kind the mean claim a, 'rnunt from each model
is approximately equal to its corresponding sample mean. The standard
232
deviations of the models are greater than their corresponding sample
values as we are fitting a curve with an infinite range to a sample 2
with a finite range. The X statistics show that, with the exception
of 74/3rd quarter, there are no significant differences, at 0.05 level,
between the model and actual sample values. The ratios of the total
expected loss statistics to the total actual. cost of claims are larger
than their corresponding values for the truncated lognormal distribution.
For the Pareto distribution of the second kind, P2 (x ; C, A, B),
parameter C can be interpreted as the 'voluntary excess' on the policy.
Estimates of C for each sample are given in table (7.13). They vary
considerably from each period of accident to the next. The mean claim
amounts are much closer to their corresponding sample values. The
standard deviations of the models are larger than their actual values but
are smaller than their corresponding values according to the Pareto
distribution of the first kind. The X2 statistics are smaller than
their corresponding values in table (7.12) and do*not indicate any
significant differences between the model and actual sample values. The
ratios of the total expected loss statistics to the total actual cost of
claims are generally smaller than those given for Pl (x ; A, B) in table
(7.12) but are larger than their respective values for the truncated
lognormal distribution. In terms of the X2 goodness-of-fit test
statistic, P2(x ; C, A, B) has provided a more exact fit to the AD data
than Pl (x ; A, B) distribution. In fact, the fit is almost as good as
that of the truncated lognormal distribution. One obvious reason for
this is that Pl(x ; A, B) has two parameters while P2 (x ; Cl A, B) and
TLN(x, E ;. u, Q 2)
each have three parameters. Therefore, given good
computer facilities, we would prefer the truncated lognormal model, for
the distribution of large AD claim amounts, though just as good a fit may
be obtained by P2(x ; C, A, B) or even P1 (x ; A, B) distributions.
233
7.5 Conclusions
In this chapter we considered the trancated lognormal distribution
as a model for the incompl;: te distribution of accidental damage claim
amounts. The method of M' was described for the estimation of parameters
from grouped data. The model was then fitted to samples of AD data
truncated at lower tail values. It was shown that by ignoring the first
claim a'nount interval this model would satisfactorily fit the AD samples.
In terms of the Chi-square goodness-of-fit statistic the fit was as
good as that of the complete 3-parameter lognormal distribution. Because
this latter distribution is relatively simpler to use and to develop
mathematically its application, in place of the truncated lognormal
distribution, was recommended.
Next, the trincated lognormal distribution was considered as a
model for the distribution of large claim amounts. Our AD samples were
truncated at 1600 and the model was fitted to the incomplete samples.
It was shown that a very satisfactory fit is provided by the truncated
lognormal distribution. For the sake of comparison, the Pareto
distributions of the first and second kind were also fitted to the same
truncated samples. The P2 (x ; C, A, B) provided a fit as good as the
truncated lognormal distribution. It was suggested that when computing
is not a problem the latter model should be used. Otherwise P2(x ; C, A, B)
or even the Pareto distribution of the first kind may be used as models
for large AD claim an unts.
The effects of inflation on the parameters of the truncated
distribution were not studied, but because of the relationship between
the complete and truncated lognormal distributions we can easily arrive
at these effects by following the exposition of section 3.9. It may, in
fact, be shown that inflation increases parameter u but leaves a2 unchanged.
7. E Tables
34 2
T ble '(7.1)
TRUNCATED 2-PARAMETER LOGNORMAL DIS. ***
73/4TH QUARTER DATA
, ESTIMATION BY MULTINOM_CAL MAX. LIKELIHOOD METHOD :-' MEW= 4.654 SIGMA SQ. = O. B62
MEAN=-175.678 S. D. = 231.181
AMOUNT C ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A-E)* 2/E
31- 60 518, ' 522. -4, -182. 0.031 61- 90 461. 452. 9. 680, 0.179 91- 120 359. 346. 13. 1371. (1,488,
121- 150 239. 260.. -21. -2A45. 1.696 151- 180 213. 196. 17. 2813. 1.474 181- 210 148. 149. -1. -195. 0.007 211- 240 102. 115, 1-13. -2931. 1.470 241- 270 81. 90 , -9. . -2299. 00900 2.71- 300 5E3, 72. -14. -3997. .
2.722 301- 330 66. 57. 9. 2839. 1,421 331- 360 45. 46. -1. -346. 0,0'22 361- 390 39, 38. 1. 376. 0.026 391- 420 35, 31. 4. 1622. 0,516 421- 450 34. 26. 8. 3484, 2,462 451- 480 20. 22. -2, -931. 0,182 481- 510 . 29. 18 ,. 11. 5450o 6.722 511- 540 14, 15. -10 -525. 0,067 541- 570 S. 13, -5, -2777. 1,923 571- 600 9. 11, -2, -1171, 0.364 601- 700 29. 27. 2., 1301, 0.148 701- 800 18, 17. 1. . 751, 0.059 801- 900" 20, 9. 7654, 7.364 901-1000 6, E3, -2, -1901. 00500
1001-1100 4, 5. ' -1. -1050. 0.200 1101-1200 4, 4. 0. 0. 0.000 1201-1300 3. -2. -2501.
-1301-1400 3. 2. 1. 1350. 01 0.2010 " 1401-1500 1. 2. -1. -1450. 1501-1600 0. 1. -1, -1550,
1601-1700 1. 0. 0ý . 1701-1800 0. -1: -1751. 1801-1900 0, 1. -1.
-1850. 1901-2000 1, 0. 1. 1950, 2001-2100 0. 0, 0, 0. 2101-2200 0. 0. 0. 0. 2201-2300 0, 0, 0, 0. 2301-2400 1, 0, 1, 235(; ---------- ------------- ------------- ----------
, ----------
0.667 TOTAL
---------
2567,
----------- 2562.
----------- -------------
--- 3737, ------------
------- 31.809
---- DEo
--- r`4 TOTAL EXP. LOSS '
-- = n. 8 P>0.10 TOTAL ACT. COST
? 35
Table (702)
'ý# TRUNCATED 2-PARAMETER LOGNORMAL DIS. *##
' 73/4TH QUARTER DATA
ESTIMATION BY MULTINOMIAL MAX. LIKELIHOOD METHOD : - MEW= 4.633 SIGMA 'SCE .=0., 881
MEAN- 208.871 S. D. = 202.444
AMOUNT F. ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A-E)**2/E
61- 90 461. 459. 2. 151. 0.009 91- 120- 359. 348. 11. 1160.. 0.348
121- 150 239. 260. . -21,. -2845. 1.696
151-180 213. 195. 18, 2979. 1,662 181- 210. 148. 149. " -1. -195. 0.007 211- 240 102. 115. -13. -2931. 1,470' 241- 270 81. 90. -9. -2299. (1.900 271- 300 58. 71. -13. -3711. 2.380 301- 330 66. 57. 9. 2839. 1.421 331- 360 45. 46. -1, x-346, 0.022 361- 390 ,
39. 38; 1. 376. 0,026 391- 420 ' 35. ' 31. 4. 1622. 0,516 421= 450 34. 26. 8, 3484. 2.462 451- 480 20. 22. /
-2, -931. 0.182 481- 510 29. 18. 11. 5450. 6.722 51l- 540 14. 15. -1, -525. 0.067 541- 570 Be 13. -5, -2777. 1.923 571- 600 9. 11. -2. -1171. 0.364
zoo 00 01 9, 27. 2. 1301. 0,148 - 7 1 17. 1. 751. (1.059'
801- 900 20. 11. 9. 7654. 7.364 901-1000 6. 8. -2. -1901. 0.500
1001-1100 4. 5. " -1. -1050. 0.200 1101-1200 4. 4. " ' 0, 0. 0.000 1201-1300 1. 3. -2. -2501. 1301-1400 3. 2. 1, 1350. 0,200 1401-1500 1. 2. -1. -1450. , " 1501-1600. 0. 1. -1, -1550. 1601-1700 11
. 1. U. 0.
1701-1800 0. 11 -1. -1751. 1801-1900, 0.
. 1. -1. .. -1850.
1901-2000 1. 0. 1". 1950. 2001-2100 0. 0, 0. 0. 2101-2200 0. 0, ne. (0, 2201-2300 0. 0. 0. 0. 2301-24011.
------ ------- 0.
- . 1,
--------- . 2350. 0,667
' ---- TOTAL '.
-----------
---- 2049. '
-----------
--------- 2047.
----------
- -
-----------
---------- 3631.
-----------
---------- 51.17
--------- D. F. '23
TOTAL EXP. LOSS --------------- 0.9 9, P >0.10 TCOTAL ACT. CAST
236
Table (703).
*#}- TRUNCATE: [? 2-PMAAMETEH LOGNORMAL DIS.
73 /a H (-WARTER DATA t
ESTIMATION BY MIJLTINUMIAL MAX. LIKELIHOOD METHOD MEW- G; 64B : iý. (iMA ; i[, 1. a (1 . E17(1 -
MEAN= 2[t7.6[tU S. U. = 1El6.312
AMOUNT £ ACTUAL. EXPECTED ACTUAL-' EXPECTED C[., NO. CL. NO, EXPECTED LOSS '(A-E)**2/E
91'- 120 359. 3a5. 14. 1477. 0,56- " 121- 150 239. 259,1 -2.11. -271[1. 1.544 151- 180' 2.13 . 195. 1S, 2979.1 1.662 181- 210 148. 149. -1. -195. (1.01)7 21 1- 240 102.. 115. -1.3, --2931. 1,. 47(1 24 1- 270 R1.
-? 299. [! . 900 271- 300 . 5R. 71, -13. -3711. .2 , 380 301- 330 57. 21139. 1.421
_ 331- 360 a5., 116. -1 -3a6. 0.0122 361- 90 39. 3R. 1 376. (1.026 391- 11,21) 35. i1. 4. 1622. (I 11 2 1121- 45(1 {c1. 26. 3484. . 2 . aF, 2 451-. 480 2.0. 22. -2. -931. 0,1 R2. 481- 510 29. 18. 11, 5450.. 6.722 511 560 14. 15.
. -1. -525. 0.067 541- 57() 8. 13. -5. -2777. 1 "923 571- 600 9, '
. 11, -2. ' -1171, 0.364
601- 7(1(1 29. 27. 2. 1301. (1.148 800. 1A. ' 17. 1.. 751. (1. (159
801- 9(10 20. 11 .. ' 90 7654. 7. '364 9(11-10(º(t 6. 8. -2. -1901 " 0.500
1(1 [t1-11(! 0 4. .. `_i" '-1. . -1050. 0.200
11(11-12(1(1 4. a" o" 0". 01000 1 eil 1-1 sui '. '. -7.. -25{11,
1tºCO 3. 2. 1. 135C1. Ooc00 1401-15(10 1. 2. 150 1-1161111 (1. 1, -1. -1550. 16(; 1-17O0 1. 1. 0. U. 1 1U 1-10U(1 0,
-1751. 10(11-190f1 (I. 1" -1 . -1A5o. 19o1-20(1(1 1,
21110 1-2100 2101-22.00 U. U. U. o. 2201-4 3011 0. 0. (i. U. 2301-24on
---------- I.
------------ 0.
---------- 1
---------- 2350.
--------- 0.667
TOTAL ------ ----
1500. ------------
1504,
---------- ----------
- 3932. ----------
----------- 3 ?. -
------- T
U. F. =22- TOTAL EXP, LOSS
------ --------- I. o > 0.05 TOTAL. ACT. COST
237
Table (7.4)
W, estimates of u for complete and truncated AD samples
Period of accident
Complete sample
E=30 E=60 E=90
73/4th quarter 4.516 4.654 4.633 4.648 74/1st " 4.509 4.624 4.581 4.296 74/2nd " 4.546 4.685 4.592 4.604 74/3rd " 4.637 4.765 4.758 4.581 74/4th " 4.672 4.807 4.823 4.851
75/1st " 4.684 4.821 4.881 4.891 75/2nd " 4.701 4.815 4.849 4.830
Table (7.5)
MNL estimates of a2 for complete and truncated AD samples
Period of accident
Complete sample
E=30 E=60 E=90
73/4th quarter 1.055 0.862 0.881 0.870 74/1st " 1.057 0.896 0.935 1.142 74/2nd " 1.013 0.815 0.901 0.892 74/3rd " 1.011 0.821 0.828 0.960
74/4th " 1.056 0.851 0.836 0.813 75/1st " 1.024 0.815 0.754 0.747 75/2nd " 1.056 0.880 0.846 0.861
238
Table (7.6)
X2 statistic for complete and truncated AD samples*
Period of accident
Complete sample
E= 30 E= 60 E= 90
73/4th quarter 53.4 (25) 31.8 (24) 31.3 (23) 31.3 (22)
74/1st " 56.9 (24) 49.1 (23) 47.8 (22) 38.9 (21)
74/2nd 51.8 (24) 26.3 (22) 22.8 (22) 23.0 (21)
74/3rd 69.3 (26) 41.6 (24) 41.6 (23) 35.8 (23)
74/4th " 56.4 (27) 21.6 (25) 22.0 (24) 22.0 (23)
75/1st 59.7 (26) 23.3 (24) 21.9 (23) 22.9 (22)
75/2nd " 44.6 (27) 24.7 (25) 25.0 (24) 24.1 (23)
* the number in () is the degrees of freedom
Table (7.7)
Ratio of the total expected loss statistic, T, to the total actual cost of claims for complete and truncated AD
samples (in percentage)
Period of accident
Complete sample
E= 30 E= 60 E= 90
73/4th quarter - 1.5 0.8 0.9 1.0
74/1st " 0.6 1.5 1.8 1.2
74/2nd - 1.0 1.1 0.6 0.7
74/3rd " - 1.0 1.1 1.3 0.7
74/4th " - 1.9 0.6 0.4 . 0.6
75/1st " - 2.1 0.5 0.6 0.9
75/2nd -'0.9 0.7 0.4 0.7
239
Table (7,8.1)
* TRUNCAT ED 2-PARAME TER LOGNOR MAL DIS.
73/4TH'UU ARTER DATA
ESTIMATION BY MULTINOMIAL MAX. LIKE LIHOOD METHOD : - MEW= 5.827 S IGMA SQ. - 0.3 50
MEAN= 856. 558 "s . D. - 490.8n8
AMOUNT £ ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A_E)**2/E
601- 700 29. " 30. -1. -651. 0.033 701- 800 18, " 20. -2. -1501. 0.200
" 801- 900 20. 13. 7. 5953. 3.769 901-1000 6. 8. -2. -1901. 0.500
1001-1100 4. 6. -2. -2101. 0.667 1 101-1200 4., 4.. 0. 0. 1201-1300 1. 3. -2. -2501. 1301-1400 3. 2. 1. 1350. 0.571
1401-1500 1. 1. 0. 0. 1501-1600 0. 1. -1. , -1550. 1 1601-1700 1. 1. 0. 0. 1701-1800 0. 0. 0. 0. 1801-1900 0. 0. 0. 0 1901-2000 1. 0. ' 1. 1950. 2001-2100 0. 0.. 0. 0. 2101-2200 0. 0. 0.
. 0.
2201-2300 0, 0. 0. 0. 2301-2400, 1. 0, 1. 2350. 0,8C0
TDTAL 89. A9. ------ --"------ 1400. --
ýý- 6.
---------, ----------- ------------ ------------------------r------- D. F. -4 TOTAL EXP. 'LOSS
----- ----------= 1.8 ry, P >0110
TOTAL ACT. COST
240
"1 ,
Table (708.2)
## TRUNCAT ED 2-PARAMETER LOGNORMAL DIS.
74/15T QUARTER DATA
ESTIMATION BY MULTINOMIAL MAX. LIK ELIHOOD ME THOD : - MEVr= 6.385 SIGMA SO. = D. 130
MEAN= 816. 831 S. D. - 723. 298
AMOUNT £ ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO,
" CL. NO. EXPECTED LOSS (A-E)**2/E
601- 700 26. 27. -1. -651, 0.037 701- 800 21 ,, 19: -2. 1501. 0,21 1
'801- goo 11. 13. -2. -1701, 0.308 901-1000 10. 0, 2. 1901 , (1.500
1001-1 100 5. 5. 0. 01, 1" 0: 000 1101-1200 2. 3. -1. -1150, 1201-1300 1" 2.
-1 . -1250. 1301-1400 2. 1. 1. 1350. 1401-1500 0"" 1. -1. -1450.
11501-1600 0. 0. 0. 0, 1601-1700 0. 0. 0. 0, 1701-1000 1. 0. 1. 1751. "0,1.3 -----
TOTAL ----------
79"
------------ 79.
------------
---
--------
---------- 300.
-----------
------------ ?. tja
----------- D. F. =3 TOTAL EXP. LOSS
= 0.5 7, > 0' 50 TOTAL ACT. COST .
241
Table (7.8.3)
TRUNCATED 2-PARAMETER LOGNORMAL DIS. "#*
74/2ND QUARTER DATA
EST IMATION BY MULTINOMIAL MAX. LIKELIHOOD METHOD : - MEW- 5.265 SIGMA SQ. = 0 .41 9
MEAN- 804. 178 S. D. - 368.034
AMOUNT f 'ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A=E)**2/E
601- 700 30. 29. 1. - 651. 0.034 701- 800 13.. 16. -3. -2251. 0.563 "801-..
900 11. 9. 2. 1)11. 0.444 901-1000 4.. 5. -0 6, -951. 0.200
1001-1100' 7. 7. 3. 3. 4. 4202. 1101-1200 U. 2. -2. -2301. . 1201-1300 1. 1. 0. 0. 1301-1400 1. 1. 0. 0. 1401-1500 0. 1. -1. -1450. 1501-1600 1. 0. 1, 1550. 1601-1700 0. 0. a. 0. 1701-1800 0. 0. '"0. 0. 1801-1900
----- 1.
----------- 0.
---------- 1.
- 1850. 1.125
----- TOTAL 69. -- 67. ------- ---------------
3001. ------- 2.356
--" -------- ------------------------------ -----"----------------
D. F. 12 2 TOTAL EXP. LOSS
-- . 5.4 %" P =y 3Q
TOTAL ACT. COST
242
Table (70804)
*# TRUNCAT ED 2-PARAMETER LOGNOR MAL DIS.
74/3RD QUARTER DATA
ESTIMATION'SY MULTINOMIAL MAX. LIKE LIHOOD METHOD : - MEW- 5.596 S IGMA SQ. = 0.3 99
MEAN= 842. 542 S . D. - 42.1.929
AMOUNT C ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A-E)**2/E
601- 700 '
32. 39. -7. -4553. 1.256 701- 600. 34. 24. -' 10. 7505. 4.167 801- 900 17. 15. 2. 1701. 11.267 901-1000 4. 10. -6. ' -5703. 3.600
1001-1100 9. 6. '3. 3151. 1.500. 1101-1200 4. 4. 0. 0, 1201-1300 0, 3. -3. -3751. 1,286 1301-1400 1. 2. '-1. -1350. 1401-1500 3. 1.. 2. 2901. 0 1501-1600 0. 1 1. -1. -1550. 1601-1700 ' 1. 1. 0. 0. 1701-1800 0.1 0. 0. 0,, 1601-1900 0. 0. 0. 0. 1901-2000 0. 0. 0. 0. 2001-2100 0. 0. 0. 0. - 2101-2200 1. 0. 1. 2150. ' 2201-2300 0.
. 0. 0. 0. 2301-2400 0. 0. 0. " 0o' 2401-2500 0. 0. 0. 0. 2501-2600
---------- 1.
------------ 0.
------------ 1.
--------- 2550. 01800
TOTAL
---------- 107.
------------ 106.
----------- -
--------- -
------------- 3050,
------------- ------- 12.876 -------
TOTAL EXP. LOSS DF
-- TOTAL . COST
3.4 P = 001 v01
243
k
Table (7080 )
*# TRUNCATED 2-PARAMETER LOGNORMAL-DIS.
74/4TH QUARTER-DATA
ESTIMATION BY MULTINOMIAL MAX.. LIKELIHOOD METHOD MEW= 6.183 SIGMA SQ. = 0.223
MEAN- 845.861 S. D. = 634.330
AMOUNT £ ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO, EXPECTED LOSS (A-E)**2/E. - 601- 700 42. - 43. -1. -651. .
0.023 701- Boo 28. 29. -1, -751. 0.034 801- 900 20. 20.
_ 1 0. . 0. , 0.000
901-1000 17. 13. 4. 3802. 1.231 1001-1100 8. 8. 0, 0. 0.000
. 1101-1200 5. 5. 0, 0. 0,000 1201-1300 " 1. t 4. -3. -3751. 1301-1400 5. 2. 3,. 4051. 01000 1401-1500 0. 2. -2. -2901. 1501_1600 1. 1, 0. 0", 1601-1700 00' 1, -1, -1650. 1701-1800 0. 0. 0. 0. 1001-1900 0. 0. 0. 0. 1901-2000 0. 0. 0, 0. 2001-2100 1. 0. 1. - 2050. 2101-2200 0. 04 0. 0. 2201-2300 0.1 0.. 0. 0. 2301-2400 1. 0. 1. 2350, 0.200
TOTAL
----------
129.
------------
128,
---------- ---------
- --2550. -
-----------
-----1,539
--- -------- D. r" =5 TOTAL EXP. LOSS
- ---= 2.3 %
r>0.90
TOTAL ACT, COST
244
Table (7.8.6)
*# TRUNCATED 2-PAFIAMETER LOGNORMAL DIS. #**
75/1ST QUARTER DATA
ESTIMATION BY MULTINOMIAL MAX. LIKELIHOOD METHOD : - ME'W= 1.278 SIGMA SQ. - 1.614
MEAN= 834.771 S. D. = 15.667
AMOUNT E ACTUAL- EXPECTED ACTUAL- EXPECTED CL. NO.. CL. NO. EXPECTED LOSS (A-E)**2/E
601- 700 39. 40. -1. -651. 0,025 701- 800- 19. 22. -3. -2251. 0.409 a01- 900 " 18. 12. 6. 5103. 3.000 901-1000" 12. 8. 4. 3802. 2.000
1001-1100 3* ,
5. -2, -2101. 0. e00 1101-1200 1.1 , 3. -2. -2301.
1201-1300 1. 2. ' -1. -1250.. .ý ooo 1301-1400 0. 2. -2. -2701. . 1401-1500 1. 1. 0. 011 1501-1600 0.
-1550. 1601-1700 1. 1ý 0. 0ý 1701-1Ei00 0. ' 0. 0, 0. 1801-1900 0. 0. 01 0. 1901-2000 0. 0. 0, 01 2001-2100 1. 0.
"1. 205.0. 2101-2200 1. 0. , 1. 2150. 2201-2300 0. 0. 0, 0, 2301-2400 0. 0. 01 0. 2401-2500 0. 0. 01 0. 2501-2600 0. 0. 0. 01 2601-2700 _ 0. 0. 01 0. 2701-2600 0. 0; 01 0. 2801-2900 0. 0. 0. 0. 2901-3000 0. 0. 0. 00 3001-3100 01 01 o0
, o. 3101-3200 0. 0.
" 0. 01
3201-3300 0. 01 01 01 3301-3400 0. 0. n. 0 3401-3500 0. 0. 01 . 01 3501-3600 1. 0. " 1,. 3550. 0.000
TOTAL
-----------
98. ----------
97. ----------- -----------
3850. -------------
890; 4 -----
TfiTA1 r'XP I nqn D'F' m4
---------------= 4.7 TOTAL ACT. COST > 0D05
245
'Fable (7.8.7)
TRUNCAT ED 2-PARAMETER LOGNO RMAL DIS.
75/2ND QUARTER DAT A
ESTIMATION BY MULTINOMIAL MAX. LIK ELIHOOD METHOD : - "MEW= 5.982 SIGMA SQ. = 0.32 4
MEAN= 873.216 S. O. = 543.923
AMOUNT f. ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS ( A-E)**2/E
'. 601- 700 36. 36. 0.. 0. 0.000 701- 800 22. 24. -2. -1501. 0.167 801- 900 22. 16. 6. 5103. 2.250. 901-1000 11. 11. 0. 0. 1 0.000
1001-1100 4. A. -4. -4202. 2.000 1101-1200 3. 5., -2. -2301. 0., 800 120.1-1300 3. 4. -1. . -12511. 130,1-1400 6. 2. '4. 5402. 1,500 1401-1500
_ 2. 2. 0. 0.
1501-1600 1. 1. 0. 0. 1601-1700 0. 1. -1. -1650. 1701-1800" 1. 1. 0. 0. 1801-1900 0. 0. 0. 0. 1901-2000 1.; 0. 1. 1950. 2001-2100 0. 0. 0. 0. 2101-2200 0_0
. 0. 0. 0. 2201-2300
----------
1.
------------ 0.
----------- 1. "
---- 2250. 0,200
TOTAL 113. 111. ---- ---------------
3801. ------- 6. ß17
D. F. 5 EXP. LOSS
----- ---------- = 3.8 % P 0. -,, 3 TOTAL ACT. COST "
ti
J
246
{ Table (7.9)
PARETO DIS. RF THE 1ST KTND *** .
734TH QUARTER DATA
ESTIMATION BY MULTINOMIAL MAX. t. IKE L THn(1D METHOD i- A= 3.188
MEAN= R75. 00(l S . D. = 449.738
AMOUNT £ ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO.
. CL. NO. EXPECTED LOSS (A-E )**2 /E 601-. 700 29. 35. -6. -3903".
. 1.029 701- 800 18. 19. -1. -751. 0.. 053 801- 900 20. 11. 9. 7654. 7.364 901-1000 6. 7. -1. -951. 0.143
1001-1 100 4' 5. -1 . -1050. 0.2C0 1101-1200 4. 3. 1. 1150. 1201-1100 1. 2. -1. -1250. 0.600 1301-1400 -3. 2. 1. 1350. 1401-1500 1. 1. 0. 0. 1501-1600 0. 1. -1. -1550. 1601-17011 1.. 1", 0. 0. 1701-1800 0. 1. -1. -1751. 1801_1400 0. 0. 0. 0. 1901-2000 1.. 0. 1. 1950.1 2001-2100 0. 0. 0. o, 2101-2200 0. - 0. 0. 0. 2201-2300 0. 0. 0. o. 2301-2400 1.
------ 10.
------------ 1.
---------- 2350.
------- 0,167
TOTAL TOTAL ----------
89. ---------
88. ------------ ----------
- 3251,
-----
------------
------ -
- D. F.
------- -5 TOTAL EXP. Lf) 3S
-- 4.2 % P =0 10 TOTAL ACT. COST ,
247
w
Table (7,10)
- *** PARETO DIS. OF THE 2ND KTND
73 4TH QUARTER DATA
-EST IMATION BY MULTINOMTAI. _
MAX. I. TKELIHOOD METHOD : A- 3.493 C- 27.94
' iMEAN= 869. 306 S. D. - 368.417
AMOUNT £ ACTUAL EXPECTED ACTUAL- EXPECTED, CL. Mf. Cl_. * NO. EXPECTED LOSS (A-E)**2/E
601- 700 29. " 29. 0. 0. 0.000 701- 800 18. 23. -5. . -3752. 1.087 801-. 900 i 20. 13. 7. 5953. 3.769 901-1000 6. A. -2. -1901. 0-. 500
1001-1100 4. 5., -1 ." -1050. 0,2.00 1101-1200 4. .
3. 1. 1150; 1201-1300 1. 2. -1. -1250. 0.000 1301-1400 3. 1. 2. 2701. " 1401-1'500 1 .. 1. 0. ' 0. ' 1501-1600 0. 1, -1. . -1550. 1601-1700 1. 1. " 0. 0. 1701-1800 0.
. 0. 0. 0.
1801-1ßO0 0. "0. 0. "0. 1901-2000 1.. 0. 1. 1950. 2001-2100 0. 0. 0. 0. 2101-2200 0. 0. 0. 0. 2201-23(10 0. 0. 0. 0. 2301-2400
-------
1.
-----------
0.
------------ 1.
-------- 2350.
---------------- 2.7 0
------- --- TOTAL
--------
89.
------ -----
87.
------------ -------- 4601.
--------------- ß0ß6
---=---- -- D. F. /.
TOTAL EXP. LOSS 0010 ----- ---------- 6.0 % TOTAL ACT. COST
248
lag
Table (7.11)
Actual. r: can claim amount and standard deviaticn for AD sa, Ics truncated at E= 600
Period of 73/4 74/1 74/2 74/3 74/4 75/1 75/2 accident
Mean 860.6 820.1 808.5 847.7 849.7 844.4 876.2
S. D. 286.0 199.0 223.0 288.0 256.0 373.0 287.0
Table (7.12)
Pareto distribution of the first kind fitted to the tail of the AD samples. (B = 600)
Period of accident
A Mean S. D. X2 (ll. F. ) R*
73/4th 3.188 875.0 449.7 9.2 (5) 4.2
74/1st 3.568 834.3 352.7 5.8 (4) 4.6
74/2nd 3.843 811.7 305.0 3.0 (3) 5.9
74/3rd 3.344 856.7 404.1 17.0 (6) 5.9
74/4th 3.259 866.3 427.8 9.6 (6) 3.8 75/1st 3.544 836.6 357.7 10.5 (5) 7.6
75/2nd 3.026 897.0 509.1 10.9 (6) 6.2
Table (7.13)
Pareto distribution of the second kind fitted to the tail of the AD samples (B = 600)
Period of accident
C A Mean S. D. 2
X (D. F. ) R*
73/4th quarter 28.0 3.493 869.3 368.4 8.0 (4) 6.0 74/1st tt 39.0 4.163 829.6 263.4 3.4 (3) 6.4
74/2nd " 5.4 3.918 811.7 294.2 2.5 (3) 4.2 74/3rd 43.0 3.938 848.3 291.4 5.4 (4) 3.6 74/4th " 30.4 3.612 860.8 344.2 5.2 (5) 4.9 75/1st " 11.5 3.686 835.6 330.6 8.2 (4) 5.1
75/2nd 24.5 3.260 890.7 427.5 8.5 (5) 4.4
*R is the ratio, in percentage, of the total expected loss statistic, ý', to. the total actual cost of claims.
249
CHAPTER 8
The Gaimu Distribution
8.1 Introduction
The gamma distribution is another positively skewed distribution
with a long tail which can assume a variety of shapes. It has,
therefore, been a most appropriate statistical model in applied science
work. Many sets of empirical data on positive valued random outcomes
of various random phenomena in science and engineering have been
represented by this distribution. Johnson and Kotz (1970) and Bury (1975)
give bibliographies on the applications of this model and the formers'
list includes references to the actuarial field as well.
Beard (1978) argues that "the amount of damage (physical) will
arise from impact and thus be related to the energy involved, that is,
to the square* of the velocity". If X is the amount of loss, and in and V
are the mass and the velocity of the car at the time of impact,
respectively, then the above statement means that
12 X is proportional to 2 mV
and hence Z=/ is proportional to V (8.1-2)
Beard (1978) then goes on to say that "velocity should have a reasonable
shape" and in Beard (1978, personal communication) suggests the gamma
distribution as a model for the distribution of velocity and hence of
the square root of the claim amount. Therefore, the problem we shall
consider in this chapter is- the application of the tti, -o-parameter gamma
distribution as a nadel for the square root of the claim amount. It
is immediately appreciated that data on the square root of the claim
amount, and not on the claim. amount itself, are required. Mien a
250
grouped frequency distribucion of the claim amount is available, for
instance like our AD data of tables (1.1) to (1.7), then the square roots
of the boundaries of the claim amount intervals may be used to obtain a
frequency distribution for the square root of the claim amount. This
is, howetier, not satisfactory since the resulting intervals will be-of
different lengths and the useful assumption which is cften made about
the concentration of observations (claims) within each interval at the
mid-point of that interval is unlikely to hold in reality. -
Therefore the insurance company provided us with data on the square root
of claim ar, unts. The claims are for accidental damage and are from
the same portfolio of corprehensively insured private rotor cars which
we mentioned in chapter 1. Data combined for all age groups, vehicle
groups, districts and types of use were made available. Six samples
relating to six periods of accident were provided. Each period consists
of one quarter of the calendar year and all claims which occured during
that particular quarter constitute a sample. The six periods of accident
were 1976/3rd quarter to 1977/4th quarter. For each sample the number
of claims grouped according to the square root of their size, i. e. Z
as in (8.1-2), were provided for z up to 38. The grouping was in equal
bands of length 2. Such a format for a particular sample would look
like the following histogram
square root of the claL, i amount
up to 2 2.01 -4 4.01 -6
number of claims
nl n2
n3
....... .
..... ...
36.01 - 33
..
n1.9
251
For larger claims details of individual claims were provided. We grouped
these into zurther bands of length 2 until the band containing the
largest claim (or claims) was reached. Because the square root of claim
amount is not readily understood in monetary terms we squared the boun-
daries of each interval for presentation of the data. The frequency
distributions of the six samples of AD data can, therefore, be seen
as the first two columns of tables (8.1.1) to (8.1.6). Both open and
settled claims have been included in each sa-nple. For 1976/3rd quarter
to 1977/2nd quarter the data were collected more than six months after
the end of each period of accident and hence no more, or only few,
claims are still to be reported. For 1977/3rd and 1977/4th quarters,
as we can see from tables (8.1.5) and (8.1.6), the total number of claims
are less than for other periods. This is because for these two periods
of accident the data were collected, respectively, three months after
the end and at the end of the periods. Therefore many more claims
are still expected to be reported to complete the histogram in each
case. We will not make any allowances for IBNR claims of these two
periods but will judge the results of the model fitting in the light of
the incompleteness of the data.
Initially, *in this chapter, we will define the gannna distribution
and will mention some of its properties. The estimation problem will
be discussed next and the method of multincnial naxinwn likelihood Gm)
will be described for estimation of the parameters from grouped data.
This method will then be used to fit the garmra distribution to the AD
square root data mentioned above. The results will be analysed and the
goodness-of-fit tests will indicate the adequacy o, f. this model. The
effects of inflation on the parameters of the model will be considered
in section 8.5. These will then be used to . redid the future distribution
of the square root of claim anuunts from past data. The actual and
252
predicted distributions will be compared, The findings of this
chapter will be summarized in section 8.7. All the tables are presenteJ
in section 8.8.
8.2 Definition and Some Properties of the Ganmia Distribution
A random variable Z is said to have a (two-parameter) gamma
distribution if its probability density function, (p. d. f. ), which we
denote by fG(z ; a, $) , is of the form :
za-1e-z/O f (Z ; 0110 (8.2-1) G $aF (a)
a -and ß are the shape and scale parameters respectively. r(a) is
called "the gam a function" and is defined by
r(a) - Jt'c -t dt (8.2-2) 0
which, by integrating by parts, can be shokn to be
r (a) - (a-1) r (a-1) (for a> 0) (8.2-3)
V, hen a is a positive integer r (a) is the factorial function, i. e.,
T (a) = (a-1) (8.2-4)
If in (8.2-1) we put a= 1 then fG(z ; a, i) will be the p. d. f.
of the standard gamma distribution.
For a=B=1. (8.2-1) will give the p. d. f. of the exponential
distribution.
I£ 2 and a-2 then (8.2-1) will be the p. d. f. of the Chi-square
distribution with v degrees of freedom.
If we make a transformation, on the random variable Z, of the form
z ->. z-c which is a translation along the --axis then (8.2-I) will. be the
p. d. f. of the 3-parameter gam m distribution whose properties can be
253
easily derived from those of f0(z ; a, O). Therefore it will suffice
to consider only the 2-parameter ganmra distribution here. Further
reference can be made to Bury (1975) and Johnson and Kotz (1970).
We denote the distribution function of fG(z a, ß) by G(z
For the standard gamma distribution we have
z 1
G(z ; al1) =räJz lez dz
0
(8.2-5)
where the integral is called "the incomplete gamma function" and is
denoted by rZ (a) . Therefore
z(a) (8.2-6) G(z ; a, 1) r
r (a)
Since ß is a scale parameter we can show that
G(z ; a, $) = G(ß ; a, 1) (8.2-7)
hence the two parameter ganra distribution function is
rz/ß(°`) G (z ; a, ) = (8.2-8)
r (a)
Let x2 v(u) denote the distribution function of a Chi-square random
variable, U, with v degrees of freedom. As we mentioned earlier,
fG (z ; a, 2) is the p. d. £. of a Chi-square random variable with v= 2a
degrees of freedom. Hence the relationship between the gaim a and the
Chi-square distribution functions will be
2 G(z ; a, ß) = G( Zß
; a, 2) X2a (2ß ) (8.2-9)
Therefore tables of the Chi-square ctnnulative distribution function may
be used to calculate G(z ; a, ß). Pearson (1922) has, however, tabulated
G(z 1), to seven decimal places, for a=0(. OS)1(0.1)5(0.2)51.
more accurate tabulation. s, of G( z; a, l), to nine decimal places, have
2-54
been given by Harter (1564) fora = 0.5 (0.5) 75 (1)165. Pearson (1922)
has shown t; cat,
G(z ; ct, ß)=e O
ýY +J/, 4 («+J+1) l
J (8.2-10)
where y>0
For computing purposes, a good approximation to the value of G(z ; a, 0)
can be obtained by taking sufficient terms in the above series. We
shall use this formula in our estimation programs and other calculations
involving the gana cumulative distribution function. The values of
the complete gamra function r(. ) , as required in (8.2-10) are computed
by a special routine which exists in the NAG1 library of computer routines.
This routine is based on the Chebyshev expansion of the complete garra
function which is given in Abramowitz and Stegun (1968).
The rth moment of Z about zero can be directly obtained from (8.2-1)
and (3.2-2) as :
E (Zr) w ßr tr a+r (8.2-11)
hence the expected value of Z is
F(Z) = as (8.2-12)
and the variance of Z can be show i tu be 2
Var(Z) = aß
The coefficient of variation of Z is, therefore, equal to a-' which is
positive and depends en the shape parasieter only. The 3rd and 4th central
nornents of Z, denoted by a and 04 respectively, can be shown, by using
(8.2-11) , to be 3
63 = 2aa (8 2-14)
and 84 = 3aß 4
(rß+2) (5.2-15
1- Nottin harn Algorithmic Group L5
The coefficient of skewness, yl , is, therefore,
yl = 2a >0 (8.2-16)
hence all gamma models are positively skewed. The coefficient of
kurtosis is y2 where
y2 =3+ 6a-1 >3 (8.2-17)
which implies that all gamma densities are more peaked than normal
densities.
For large values of a, yl and y2*approach their normal values 0 and 3
respectively. Thus for large a the central portion of the corresponding
gamma p. d. f. should be well approximated by a normal p. d. f.
lVhhen a ,l the gamma distribution has a single mode which is located at
Zmode =ß (a-1) (8.2-18)
For values of a<1 the gamma p. d. f. has a singularity at the origin
and fG(z ; a, R) -} °° as z -'- 0.
The median, and in general the quintile of order q, of the gamma
distribution cannot be expressed in terms of its parameters but may be
obtained, by interpolation, from tables of the gamma distribution function.
As we mentioned in section 8.1 we intend to use the gamma distribution
as a model for the distribution of Z, the square root of claim amount X.
The mean and standard deviation of Z do not have any significance in
monetary terms. Therefore, it is important to obtain the mean and
standard deviation of X in terms of a and a, the parameters of the
gamma model. The expected value of X is
1)
E (X) = E(Z). since Z=
but from (8.2-11) and (8.2-3) it follows that
E (x) r (a+1) (r . 2-19)
256
The variance of X is by definition
2242
Var (X) = E(X) - (E "X)) z- E (Z )- (11(X) )
On using (8.2-11), (8.2-3) and (8.2-19), respectively, it follows that
42 6 ar(X) =ßa (4a + 10 +ä) (8.2-20) V
8.3 Estimation of the Parameters
Johnson and Kotz (1970) deal with the estimation problem when values
of individual observations in the sample are known. They consider the
usual methods of moments and maxi= likelihood. We are, however,
interested in estimation from grouped data. Therefore, in this section,
we will consider the method of multinormal maximum likelihood GM).
Let us assume that we have a sample of grouped data where n independent
random observations (claims) on a random variable Z (in our case, the
square root of the claim anwunt X) have been grouped according to their
size into k mutually exclusive intervals. Further, let ni be the number
of observations (claims) in the class interval (zi^1, zi), for
i=1,2, ... ,k, such that k
n= ni i=1
If Z has a gamma distribution given by G(z ; a, $) in (8.2-7), then let
pi be the probability that an observation (claim) occurs in the interval
(z. -. 1, zi), i. e.
ý pi = G(zi ; x, ý)
For i= is taking zo = 0,
pl = G(z1 ; a, a)
- G(zi-1 ' a, ß) (8.3-1)
for i=2,3, ... ,k
(8.3-2)
Therefore, the sariple 1.; kelihood function will be proportional to L
257
where 1. nl
" (8.3-3) I. = P1 i=1
and the loglikelihood function will be
k log L= nl log pi (8.3-4)
: i=1
where pi is as defined above. The values of G(zi ß) are calculated
from (8.2-10). AA
The maximum likelihood estimates of the parameters, (a, ß), are obtained
by maximizing log L simultaneously with respect to a and ß. To avoid
solving non-l: noar equations (which is a laborious task when k, the
rjwnber of intervals, is large) we suggest the confuting technique
described in section 3.7.7 for maximizing log L directly with respect
to the unl<no;, m parameters. To start the iteration process we can use
the moment estimates of a and ß. For grouped data the sample moments
can be calculated by assuming that in each interval (zi_1, zi), -all the
nl observations are concentrated at the mid-point of the interval, i. e.,
at (zi_1 + zi)/2. The sample first and second n ments when calculated
can be put equal to their corresponding population values to yield the
moment estimates of a and g.
8.4 Application of the Camera Distribution to the AD'Sguare'Rdot Data
To examine the adequacy of the garna distribution as a model for
the distribution of the square root of the AD claim amount we wrote
computer progr=, `P? for estimating parameters a and ß from a swrple 1
of grouped data. The hu". method described in section 8.3 was used.
The sw'nples of the square root. AD data, which were described in section
8.1, were input to the prograi<<. For each saiiple the program colrputed
and printed cut the N NI estimztes of a and 0, the shape and scale
2058
paraneters respectively , of the gamma distribution. The mean and
standard deviation of the claim amounts according to the fitted model
were also produced. The program also provided an extensive table of
results, giving in particular the values of the expected claim numbers,
the expected loss and the contribution to the total X statistic for W.
each claim amount interval. The results for the square root AD data
are presented in tables (8.1.1) to (8.1.6). The original grouping of
the data was in bands of length 2 but, for convenience in interpretation,
we have squared the boundaries of each interval and, therefore, the
grouping in the tables is in terms of the amount of claim and not its
square root. From these tables we can see that a is greater than 1
for all samples. Hence the model portrays the mode of the sample
histogram. The actual minus expected claim numbers are not small and
in few of the intervals the contributions to the X2 statistic are large.
The Xi statistics are significant, at 0.05 level, for all the samples.
These indicate significant differences between the model and actual
sample values.
The expected loss statistic for each interval was calculated by multiplying
the actual minus expected claim number by the square of the mid-point
of the original interval in terms of Z, the square root of the claim
amount. The values of this statistic show considerable disagreement, in
monetary terns, in the middle range of the claim account distribution.
Our analysis shows that the 2-parameter gamma distribution is not an
appropriate model for the distribution of the square root of AD claim
amounts. The fit of this model to the AD square root data is worse
than that of the 2-paramoter lognormal distribution to the AD data. Fence it
seers that collection of data on the square root of the claim amount is an
unnecessary task. To ponder on i: lry the fit. of the nedel should be so
259
poor we should examine the two assumptions which we made in section 8.1.
These were :
(i) - Z, the square root of the claim amount X, is proportional
to the velocity of the car at the time of impact (since X 2
is proportional to 7 mV , the kinetic energy of the car
at the time of impact).
(ii) - The velocity of the car at the time of impact has a gania
distribution
As regards assumption (i), it may be more reasonable to assume that
the claim amount X is proportional to mV, the momentum of the car at
the tijre of impact, and not to its kinetic energy. Hence, if assumption
(ii) holds, we should expect the claim amount X to have a gamma
distribution. To examine this on our AD data samples, presented in tables
(1.1) to (1.7), we used the method of moments to estimate the parameters
a and ß. -Th. - estimates of a were less than 1 for each sample, hence
indicating that the garnna model had no node to portray that of the
histogram of the sample values. The model also had a much shorter tail.
than the sample histogram. The values of the X2 statistic were
calculated from each sample and its corresponding gamma model.. These
were consistently large and indicated significant differences between
the model and actual sample values. Again the fit was considerably
poorer than that of the 2-parameter lognormal. This is not totally Un-
expected since the lognormal model has a mode and a longer tail than
the gaiima and is, therefore, more capable of representing the mode
and the long tail of the claim amount histogram.
There are no statistics available on the velocity of cars at the
time of impact to allow the examination of assumption (ii). Therefore,
even if assumption (i) or its revised form (X proportional to rr%o is
260
considered reasonable, there is no reason why Z, the square root of
claim wount X, (or X itself) should have a gamma distribution since it
is possible for V to have some other distribution.
8.5 The'Effects of Inflation on the Parameters of the Model
Although the gamma model proved not to be appropriate for the
distribution of the square root of the AD claim amounts, however, to
make this chapter complete, we consider the effects of inflation on
the parameters of the model. We follow the exposition of section 3.9
and investigate these effects theoretically. As was mentioned in that
section, we would need to allow for such effects when a prediction of
the future distribution of the square root of claim amounts is required.
Let us, therefore, again assume that the effect of inflation is to
increase a claim of amount X to U= X(1 + i) over a period of time where
i is the effective rate of inflation, according to some suitable index
of wages or prices, for that period. If Z- (Thas a gamma distribution
with p. d. f. fG(z ; a, a), as given by (8.2-1), then by a transformation
of variables it can be shown that
= rZ where r= (1 + i)
will have a gap-va distribution with parameters a and rß . Therefore
inflation affects ß and leaves a, the shape parameter, unchanged.
This means that when a gamma model is found appropriate for the distribution
of Z, the square root of the claim amount from a particular class of
insurance business, we can "update" the scale parameter ß, with
respect to inflation, and hence obtain the distribution of Z in a
future period.
8.6 Prediction for theAD Square Root Data
261
For the purpose of demonstrating our prediction technique for
the distribution of square roots of claim amounts ( despite the inadequacy
of the gonna model for the AD data) we wrote computer program P26. The
WL estü. ates a and for a particular period of accident and the rate
of inflation, calculated from the General Index of Retail Prices, are
input to the program. The parameters of the gamma model for the
future period are then calculated and the program prints out an extensive
table of results for comparing the actual and predicted distributions
of the square root of the claim amount. To see the result of predicting
one year ahead we used 1976/3rd and 1976/4th quarters to predict the
distributions for 1977/3rd and 1977/4th quarters respectively. The
results are presented in tables (8.2.1) and (8.2.2). The tables are
self-explanatory, and wire can see that the actual minus expected number
of claims is large for some intervals and that the Chi-square
statistics are very large and indicate highly significant differences
between the predicted and actual sample values. We did not expect to
obtain satisfactory results, since the fit of the model to the actual
data of 1976/3rd and 4th quarters was so poor in both cases.
8.7 Conclusicns
As we mentioned in the introduction to this chapter, encouraged
by _a
remark by Beard (1978), we studied the g an, ma distribution as a
model for the distribution of the square root of the claim =aunt. The
properties of the ¬aru a distribution were studied and the bM method
was considered for estimation of its parameters from grouped data. In
order to show the application of this mod. el in practice, we obtained
six samples of the square root of AD claim amounts. The model was fitted
to each sample by ML method and in each case it exhibited a sharper peal:
262
and a shorter tail than the histogram of the sample values. The Chi-
scviare goodness-of-fit test statistic indicated highly significant
differences between the model and actual sample values. Judged by
this statistic, the fit in each case was worse than that of the 2-
parameter lognormal model to the AD claim amounts as shown in chapter 3.
In section 8.4, based on the assumption that the amount of claim should
be proportional to the momentum of the car at the time of impact, we
examined the ganuna distribution as a model for the distribution of the
claim amount itself. Hence we fitted this model to the AD samples of
tables (1.1) to (1.7). The values of a, the shape parameter, were less
than 1 in each case. This indicated that the model ignored the distinct
mode of the actual distribution of claim amounts. The model also
exhibited a much shorter tail than the histograms of the samples. Judging
2 b) ,X statistic, a very poor fit was obtained. In each case, the fit
was worse than that of the 2-parameter lcgnonnal distribution as shown
in chapter 3. This was not surprising since the lognormal distribution
has a longer tail than the gamma and also always possesses a mode.
For the sake of a more complete study, the effects of inflation
on the parameters of the gamma model, when used for the distribution
of the square root of the claim amount, were considered. It was shown
that inflation increases ß, the scale parameter, but leaves a, the
shape parameter, unchanged. Predictions were made, using two samples
of our square root AD data and, as expected fr ., the poor fit of the
gamma model to the sample values, the predicted distributions were sho., n,
by the X2 statistic, to differ significantly from the actual ones.
Wo therefore ro j cct the gazitra distribution as ai cde1 for the
distribution of JD claim amounts or the square roots of clam vnounts.
For the latter, we b�lieve that the tasr, of collecting data on the square
263
root of claim amount is unnecessary when more adequate models, like
the lognor 1, exist for the distribution of claim. amount itself.
8.8 Tables
264
"2b1e i ß, 1.1i
* TWD-PARAMETER GAMMA DIS. *#*
76/3RD QUARTER DATA
ESTIMATION By MULTINOMIAL MAX. LIKELIH000 METHOD : - ALPHA - 3.474 BETA=. 3.866
MEAN=-232.223 S. D. = 262.746'
AMOUNT £ ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO, CL. N0. EXPECTED LOSS,. (A-E)**2/E
UPTO 4 14. 13. 11 1, 0,077 4- 16 90, 116. 4, 36. 0,186
16- 36 147. 178. -31. -777. 5.399 36- 64 240. 243. -3. -147. (1,037 64- 100 297. 271. 26. 2108, 2.494
100- 144 , 290. 265. 25. 3028, 2,358
144- 196 260. 239. 21, 3552. 1,1345 196- 256 188. 204. -16. -3602. 1.255 256- 324 175. 166�'
' 19, 2603. 0.486
324- 400 119, 130 , -11, -3973. 0.931 400- 484 76. 99. -23. -10148. 5.343 484- 576 57. 74. -17, -8997. 3.905 576- 676' 49. 54. -5. -3126. 0.463 676- 784 45. 39. 6. 4376. 0,923' 7134- 900 28.. 28, . 0, 01 00000 900-1024 21, -20, 1,
. '961. 0,050
1024-1156 18. 1.4. 4. 4357, 1.143 1156-1296 17. 9, 8. 9803. 7.111 1296-1444 4. 6. -2. -2739. 0.667 1444-1600 2. 4, -2. -3043. 1600-1764 B. 3. " 5. 84017, . 1764-1936 1. 2. -1, -1949. 1936-2'116 1, 1. 0. 2116-2304 2. 11 1. 2209. 2304-2500 2. 1, 10 2401. 2500-2704 0. 0, 0. 01 2704-2916 01 0. 01 01 2916-3136 0. 0. 00 0. 3136-3364 0. 0, 0. 01 3364-3600 0. 01 01 0. 3600-3844 0. . 0. 0, 0. 3844-4096 01 0, 01 00 4096-4356 01 0. 0, 4356-4624 01 0.. 06 0. 4624-4900 01 0, . 0. 0. 4900-5114 0. 0. 0, 0. 51Rh1.. 5476 0. ' 01 0, 0, 5 76-6776 0 1. 5626. 0 ------------ -------- Y... - ----------- ----------- --------------- ---------
TOTAL .. ---------
2152, --- . -- +----
2150. ----------- -----------
11067, -----------
36.767 -- -----------
TOTAL EXP. LOSS 2.2 as J !ýa
TOTAL ACT. COST
26
Table (8.1.2)
# TWO-PARAMETER GAMMA DIS. *#*
76/4TH RUARTER DATA
ESTIMATION BY MULTINOMIAL MAX. LIKELIHOOD METHOD t- ALPHA = 3.733 BETA= 3.888
MEAN= 266.975 - S. D. - 290.604
AMOUNT f. ACTUAL -EXPECTED ACTUAL- EXPECTED CL. NO, CL. NO. EXPECTED LOSS (A-C)**-2/E
UPTO .4 11. 10. 1. 1. 01.100
4- 16 107. 82. 25. 226. 7.622 16- 36 145. 192. -47. ' -1177. 11.505
. 36- 64 266. 287. -2.1. -1030. 1.537
. 64- 1110 . 361. 342. 19, 1541. 1.056 100- 144 361. 354. 7. 8418.. n. 138 144- 196 -359. 334. 25. 4228. 1.871 196- 256 325. 296. 29. 6529. 2.841 256- 324 252. 249. 3. 868. 0.036 324 - 400 200. 202. -2. -722, 0.020 400- 484 137. 159. . -22. -9707. 3.044
X 484- 576 112. 122. -10. -X5292. 0.820 576- 676 80. 92. ' -- 12. 4503. 1 *565 676- 784 61. " 68. -7. -5'! 05. 0.721 784- 900 51. 49. 2. 1683.. 0.082
.. 900-1024 31. 35. -4. -3845. 0.457 1024-1156 28. 25. 3, . 3268. 0.360 1156-1296 24. 18. 6. 7352. 2.000 1296-1444' 13. " 12. 1. 1369. 0.083 1444-1600 8. 8. 00 0. 0.000 1600-1764 . 4. 6. -2. -3363. 0.667 1764-1936 9. . 4. 5. 9247. 1936-21 16 1; - 3. -2. -4051. 1.286 2116-2304 2. 2. 0. 0. 2304-2500 2. 1. 1. 2401.
. 2500-2704 1. 1. 0.. . 0,
2704-2916 0. 1. -1. -2810. 2916-3136 0. 0. 0. 0. 3136-3364 01. 0. 0. 0. 3364-3600 1. 0. 1. 3482. 3000-3844 . 1. 0. 1, 3722. 3844-4096 0. 0. 0. 0, 4096-4356 0. 4356-4624 ----------
1. ---- --- ----
0. --------- ! ----
1. -----------
4490. ----------
1.800 --
ToTAL - 2954-. 2954. 6648. ------------
39,610
Q. F. -70 TOTAL EXP. LOSS . 0.8 "
TOTAL ACT. COST
26G
Table (8.1,3)
-ý^# TEND-PARAMETER GAMMA DIS. -
$**
77 f 'i S'! ' QUARTER DATA
ESTIMATION 0Y. MULTINOMIAL MAX. LIKELIH00D METHOD '; - . ALPHA = 3.822 BETA= 3.790
MEAN= 264.715 S S. D. = 284-. 505
AMOUNT ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A_E)**2/E
UPTO 4 7. 7. . 0. 0. 0.000
4- 16 85. 62. 23. 208, 8.532 16-. 36 107. 151. -44. -1102. 12,821 36- 64
0 6 2069 280
22.9.. 2
-23. -1129. 2�310 4- - 10 . 75. 5. 405, 0.091
110- 144 328, 286. 42. 5087. 6.168 144 196 299. 271. 28. 4736. 2.893 196-. 256 262. 239. 23. 5178. 2.213 256- 324 196. 201. -5. -1446. 0.124 324- 400 151. 163, -12. -4334, 0,8113 400- 484 104. 127. -23. -10148. 4.165 484- 576 79. 97. -18. -9526. 3.340 -576- 676, 62. 73. -11. . -6878. 1.658 676- 784 50. 53. -3. -2188. 0.170 784=- 900 45. 138. 7. 5889. 1.269 900-1024 27. 27. 0. 0. 0.000
1024-1156 22. 19. 3. 3268. 0.474 1156-1296 8. 13. -5. -6127. 1,923 1296-1444 16, 9. 7.
. 9586. 5.444 1444-1600 8.
. 6. 2. 3043. 0.667
1600-1764 6. 4. 2. 3363. 1764-1936 4. 3. 1. 1849. 1.286 1936-2116 5. 2. 3. 6076. 2116-2304 2. 1. 1. 2209. 2304-2500 . 0. 1. -1. -2401. 2500-2704 0. 1. -1. -2602. 2704-2916 0. 0. 0, 0. 2916-3136 1. 0. 3026. 3136-3364 0. 0. 0. 0, 3364-3600 0, 0, 3600-3844 0. 00 - 0. 00 % 3844-4096 0. 0, 0. 0, 4096-4356 -
1. ----
" 0" ------------
11 ----- w_w»
4226. _- TOTAL L. 2361. 2358. 102fýH.
- ----- K,
------ ----------- - - ' 654 - - --------- --.. _---ww_ww--__ww.. -_w_ w - ------ TOTAL EXP. LOSS
D. F. '19 --- .. 1.6%
TOTAL ACT, COST
267
Table (8.1.4)
* TWO-PARAMETER GAMMA *** DIS,
77/2ND QUARTER DAT A
ESTIMATION BY r ULTINOMIAL MAX. LIKE LIHOOD METHOD ; - ALPHA = 3.746 BETA= 3.941
MEAN= 276.19A S. D. = 300.056
AMOUNT £ ACTUAL. EXPECTED ACTUAL- EXPECTED CL. NO. - CL. NO. EXPECTED LOSS (A-E)*-K-? /E
UPTO 4 5. 6. -1.. -1. 0.167 4- 16 68. 51. 17. . 154. 5.667.
16- 36 93. '
120. -27. -676. 6.075 36- 64 162. - '182. -20. -981. 2.198 64- 100' 236. 219, 17, 1379. 1.320
100- 144 237. 229. 8. 969. 0.279 114- 196 240. 218, 22. 3721. 2.220. 196- 256 209. 195.
, 14. 3152. 1.005
256- 324 153. ,
165, . -12. -3470. 0,1373
324- 400 137.. 135. " 2. 722. 0,030 4(101- 484 99. . 107. -8. -3530. ' 0.598 484- 576 71. 83. -129 -6351. 1.735 576- 676 58. - 63. -5. -3126, 0.397 6.76- 784 51. 47. 4. " 2917. 0.340 784- 900 31. 34. -3. -2524. 0.265 900-1024 22. 25. -3. --2884. 0.360
1024-1156 14, ' 18, -4. -4357. 0.889 1156-1296 15. 13. 2. 2451. 0.308 1296-1444 10. " 9. 1. 1369. 0., 111 1444-1600 10. 6. 4. 6086. 2.667 1600-1764 2. 4, -2. -3363 1764-1936 5. . 3. 2. . 3699. ý, CCO 1936-2116 5. 2.. 3. 6076. 2116-2304
, 0. 1. -1. -2209.
2304-2500 1. 1. 0. 0. 2500-2704 2.
. 1. . 1.
. 2602.
2704-2916 n. 0. 0. 2916-3136 1. 0. 1. 3026.
- 3,200
TOTAL
---------- '1937.
------------
1937.
------------ --------- 4848,
----
-------------- X0.702_
--
--------
TOTAL EXP. LOSS D. F. -- -
' --- 019 % P <0, ' TOTAL ACT. COST
263
Table (8,1.5)
## TWO-PARAMETER GAMMA OIS. ##*
77/3RD QUARTER DATA'
ESTIMATION By MULTINOMIAL MAX. LIKELIHOOD METHOD :- ALPHA = 3.331 BETA= 4.372
MEAN= 275.774 S. D. = 319.158
AMOUNT £ ACTUAL EXPECTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A-E)**2/E
UPTO 4 9. 10. -1. -1. 0.100 4- 16 96. 59. 37. 334. 23.203
16- 36 67. 122. -55. -1378. 24.795 36- 64 159. 169.,
_ -10. -491. 0.592 64- 100 191. 192. - -1. -81. 0.005
100- 144 189. 195. -6. -727. 0.1135 144- 196 210. 182. 28. 4736. 4.308' 196- 256 174. 161, 13. 2927. 1.050 256- 324 158. 136. 22. 6362. 3.559 324- 40G 107. 112. -5'. -1806. 0.223 400-484 86. 90. -4. -1765.. 0.178 484- 576 74. ' 70" 4. 2117. 0.229
' 576- 676 56. 54. 2. 1251. 0.074 676- 784 32. 41. -9. -6563. 1.976 784- 900 21. 31. -10. -8413. 3,226 900-1024 13. 23. -10. -9613. 4.348
1024-1156 13. 17. -4. -4357. 0.941 1156-1296 11. 12. -1; -1225. 0.083 1296-14,14 14. 9. 5. 6847. 2.778 1444-1600 5. 6. -1. -1521. 0.167 1600-1764 4. 4. ' 0. 0.. 00000 1764-1936 4" 3. 1. 18gq. 1936-2116 2. 2. 0. 0. 0,200 2116-2304 2. 2. 0. 0. 2304-2500 3. 1. 2. 4803. 2500-2704 1. 1. 0. 0. 2704-2916 0. 1. -1. -2810. 2916-3136 1. 0. "1. 3026. 33136-3364 0. 0. 0. 0" "3364-3600 0. 0. 0. U. 3600-3844 0. 0. 0. 0" 3844-4096 0. 0. 0, 0. - 4096-4356 0. 0. 0. 0. 4356-4624 2. 0. 2. 8979. 3.200
TOTAL
----------
1704.
----------- 17050
------------ ----------- 24I9.
µ- -
------- 75.4+0-----
----- D. F.
------------ _- ?0
TOTAL EXP. LOSS ----" ------ = 0 "5 oil v ý"! 1:
TOTAL ACT. COST
269
Table (8,, 1.6)
*** TWO-PARAMETER GAMMA DIS. ***
77/4TH QUARTE n DATA
ESTIMATION BY MULTINOwlI AL MIAX. LIKE LIH00D METHOD ; - ALPHA 2 . 690 BETA= 5.387
MEAN= 289 . 606 S. D. = 375.684
AMOUNT E ACTUAL EXP¬CTED ACTUAL- EXPECTED CL. NO. CL. NO. EXPECTED LOSS (A-E)**2/E
UPTO 4 3. 6. -3, -3. 1.500 4- 16 .
50. 25. 25. 226. 25.000 16- 36 19. 1 . 42. ' -23. -576. 12.591 36- 64 31. 51. -20. -981. 7.843 64- 100 57. 54. 3-. 243. 0.167
100- 144 54. - 52. 2. 242. 0.077 144- 196 '(7. 48. 19. 3213. 7.521 196- 256 42. 42. 0. 0. ' . 0.000 256- 324 53. 36. 17. 4916. B. 028 324- 400 24. 30. -6. -2167. 1.200 400- 484 17. 25. -8. -3530. 2.560 454-. 5.76 13. 20. -7. -3705. 2.450 576- 676 2?. 16. 6. 3752. 2.250 676- 704 10. 12. -2. -1459'. 0.333 784- 900 9. 10. -1. -641. 0.100 9O0-1024 13. 7. ; 6. 576E+. 5.143
1024-1156 0. 6. . -6. . -6536. 6.000
1156-1296 0. 4. -4. -4901. 1296-1444 ,
0. . 3. _. -3. -4108. ' 7.0Q0
1444-1600 3. 2. 1. 1521. 1600-1764 1. 2. -1. -1601. 01000 1764-1936 5. 1. 4. 7396. 1936-2116 2. 1. 1. . 2025. 211E-2304 3. 1. 2. 4419. 2304-2500 0. 1. -1. -2401. 2500-2704 . 1. 0.. 1. 2602,
' 2704-2916 1. 0. 1" 2810. 16.000
TOTAL.
----------- 500.
---------- 497.
---------- -----------
----------------------- 0244.105.767
------------- - D. F. - ---------
= 17 TOTAL EXP. LOSS
------ -----. ---- 4.3 P <001 TOTAL ACT. COST
270
Table (8.2.1)
*** - (WO-PARAMETER GAMMA DIS. ***
PREDICTION OF 7? /3RD (LJARTER CLAIMS CO ii USING 76/3RD (QUARTER PULT. MAXL. PARAMETERS
ALPHA= 3.474 BETA= 3. A66 INFLATION RATE I 1fß. 54 CALCULATED FROM GENERAL INDEX OF RETAIL PRICES.
PREDICTION P!, HAPMETEf1S ARE ALPHA= 3.474 BETA- 4.173 MEAN CLAIM AMOUNT= 2? 0.63 5. D. =306.19
ACTUAL 7? /3HU PARAMETE RS : - ALPHA = 3.331 BETA= 4.372 MEAN CLAIM hiAOLJNT= 275.75 S. D. =319. i5
AMOUNT f, ACT . NO. E XP. NO. A-E E XP. LOSS (A-E) **2/E UPTC) 4 9. A. ' 1. 1. 0.125
4- 16 96. 55. 41. 3? 0. 30.504 16- 36 67. 119. -52. -1303. 22.723 36- 64 159. 169. -10. -491. 0.592 64- 100 191. 195. -4. -324. 0.082
100- 144 189. 199. -10. -1211. 0.503 144- 196 210. 186. 24, 4059. 3.097 196- 256 174. 164. 10. 2252. n. 610 256- 324 158. 139. 19. 5494. 2.597 324- 400 107. 113. -6. -2167. 0.319 400- 484 86. 90. -4. -1765. 0.178 484- 5? ( 74. 70. 4. 2117. 0.229 576- 676 56. 53. 3. 1876. 0.170 676- 784 32. 40. -8. -5834. 1.600 784- 900 21. 29. -8. -6730. 2.207 900-1024 13. 22. -9. -8652. 3.682
1024-1 156 13. M. -3* -3268. n. 563 1156-1296 11. 11. n. n. 0.000 1296-1444 14. 8. 6. 8216. 4.500
144d-1600 5. 6. -1. -1521. 0.167 1600-1764 4. 4. n. n. 1764-1936 4. 3. 1. 1849. 1936-2116 2. 2. n. n. 2116-2304 2. 1. 1. 2209. 2304-2500 3. 1. 2. 4803. 2500-2704 1. 1. 0. 2704-2916 0. n, n, n. 2916-3136 1. 0. 1. 3026. 3136-3366 0. 0. 0. 0. 3364-3601) 0. 0. 0. 0. 3600-3844 n. 0. n. n. 3844-4096 n. n, n. n. 4096-4356 01 0. n. 0. 4356-4624 2. n. 2. [4979.
------------ TOTAL
-----------
----------- 1704
-----------
--------- 17114 ---------
-----
-----
--------- - 119V5,
-----------
----------- 711.504
-----------
CHI 5W. STMT. = 81. g 7 n. F. = 21 F <. 01
TOTAL. ACTUAL CURT = 47220fi. TOTAL. EXPECTED (; UET t 46(I223 .
TOTAL E: Yf. LOS`* ----------------- 2.54 TOTAL ACT. COST 271
Table (8.2.2)
*** TWO-PARAMETER GAMMA DI: i. ***
PREDICTION OF 7? /4TH QUARTER CLAIM(' COST USING 76/4TH QUARTER l4ULT. MAXL. PARAMETERS
ALPHA= 3.733 (SETA= 3.888 INFLATION RATE I=13. Cä CALCULATED FROM GENERAL INDEX OF RETAIL PRICES.
PREDICTION PARAMETERS ARE : - ALPHA= 3.733 BETA- 4.133 MEAN CLAIM AMOUNT- 31)1.80 5.0. -328.50
ACTUAL 77/4TH PARAMET ERS : - ALPHA- 2.648 BETA- 5.387 MEAN CLAIM AMOUNT.. 289.54 S. a. -3? 5.62
AMOUNT F ACT. NO. * EXP. NO. A-E EXP. LOSS (A-E)**2/E n- 4 3. 1. 2. 2. 4- 16 50. 12. 38. 343. 120.333
16- 36 19. 28. -9. -225. 2.893 36- 64 31. 43. -12. -589. 3.349
-64- 100 57. 53. 4. 324. 0.302 100- 144 54. 56. -2. -242. 0.071 144- 196 67. 55. 12. 2030. 2.618 196- 256 42. 5n. -A. -1801. 1.280 256- 324 53. 44. 9. 26n3. 1.841 324- 400 24. 36. -12. -4334. 4.000 400- 484 17. 29. -12. -5295. 4.966 484- 576 13. 23. -10. -5292. 4.348 576- 676 22. 1H. 4. 2501. 0.689 676- 784 10. 14. -4. -2917. 1.143A 784- 900 9. in. -1, -841. n. IIU) 900-1024 13. A. 5. 4807. 3.125
1024-1156 n. 6. -6. -6536. 6.000 1156-1296 0. 4. -4. -4901. 1296-1444 n. 3. -3. -4108. 1444-1600 3. 2. 1. 1521. 1600-1764 11 1. Q. n.
1764-1936 5. 1. 4. 7398. 1936-2116 2. 1. 11 2025. 21 16-2304 3. (31 3. 6628. 2304-250(1 0. 0. 0. n.
2500-2704 1. 0. 1. 2602. 2704-2916 1.
------ 0.
------- 1.
-------- 281 n,
----- ----------- TOTAL.
-----------
---- 500
---------
1198
------- --------
------ -14A9 .
-----------
---------- 157.258
----------
CHI : C'I. STAT. r 191,6 , D. F. r17 P <. 01
TOTAL ACTUAL COAT Q U15477. TOTAL EXPECTED COST = 146966.
TOTAL EXP. LOIF3S
--------------- TOTAL. ACT. CO,; T
272
CHAPTER 9
Summary of Conclusions
In this work several statistical distributions were examined
as models for the distribution of claim amounts in general insurance.
A chapter was devoted to the study of each. The results of examining
the applicability of different distributions were summarized in the
concluding sections of the various chapters. In those sections,
where applicable, recommendations were made as to procedures to be
adopted when testing whether a sample can be regarded as drawn from
a particular distribution. The preferred method (methods) of
parameter estimation for the model was also suggested and its
performance on the accidental damage (AD) data was reviewed. In
addition, the effects of inflation on the parameters of the
model were summarized and, with appropriate models, the
performance of the prediction technique on the AD data was reported.
To avoid unnecessary repitition we suggest that for a sunnzry of
the findings on each model reference should be made to the
relevant conclusions at the end of each chapter. The purpose of
this chapter is to summarize those aspects of the present work
which are common to all the models considered and which may be
useful in fitting distributions to general insurance data.
Mien fitting statistical models to empirical data we have to
deal with two problems. Firstly the estimation of the parameters
of the model from the sample and secondly the testing of the
agreement between the model and actual sample values. Due to the
large number of claims, general insurance claim aýx unts data are
I sually in grouped frequency format. ' Therefore the efficient
method of rultinomial rw. Lmun likelihood ON, ) is reco : ended
for. estimation from grouped data. Without the aid of computers
273
this method is not always practicable. However, as computers
are now commonly available the M method should present no
problems when our proposed computing technique is used to find
the estimates of the parameters. Applications of this method
were demonstrated on the AD data for various distributions.
With some models, like the 3-parameter lognormal or Weibull, it
is possible to use the least squares (LS) method of estimation
which is simpler than the NIML method but less efficient. In
such circumstances our LS computing technique is recommended
in place of solving a system of non-linear equations. When no
computer is at hand one of the other simpler but less efficient
methods of estimation may be adopted.
Due to the importance of testing the agreement between a fitted
model and actual sample values we devoted Chapter 2 to the study of
some goodness-of-fit tests. The Chi-square is the one test which
is widely used by actuaries and others. As it is not possible to
interpret the value of this statistic in monetary terms,, we
recommended supplementing it by our proposed total expected loss
statistic T. The T statistic is a measure of overall agreement in
monetary terms between the model and the actual sample values.
Another goodness-of-fit test which is usually more 'powerful'
than the chi-square is the Kolmogorov-Smirnov test. This test
does not appear to have been exploited in actuarial work. It is,
therefore, dealt with in Chapter 2 where the application procedure
for its use is also described. In practice, we recommend that,
when conditions allow (see Chapter 2), all three statistics be used since
in some cases they nay tell a slightly different story. In our
analysis of the AD data whenever possible we used all these statistics
for examining the goodness-of-fit of the models to the actual
sample values . 274
One of the purposes of model fitting is the prediction of the
future distribution of claim amounts and hence the future cost of
claims to the insurer. Inflation is believed to be the main
cause of increase in the claim amount over time. Therefore, its
effects on the parameters of each model were studied. A prediction
technique was proposed which consisted of updating the parameters
of the past model with respect to a given rate of inflation, as
calculated from a suitable index of wages or prices. These
updated parameters were then used in the model for the distribution
of future claim amounts. Mhenever a suitable model was found for
the AD claim amounts this technique was deomonstrated on the past
data and shown to perform very well (i. e. in terms of the goodness-
of-fit tests, no significant differences were observed between the
predicted and actual distributions). It was also shown that for
AD claim amounts he appropriate index, for the calculation of the
rate of inflation, is the General Index of Retail Prices. The
effectiveness of our proposed total expected loss statistic, T,
is clearly demonstrated when judging the agreement between the
predicted and actual distributions since in that case it shows-by
how much, in monetary terms, we have overpredicted or underpredicted
the total cost of claims. The advantage of using a statistical
model was made apparent in section 3.14.2 by the fact that it gave
a smaller over-prediction of the total cost of claims than the
simple method of updating the sample mean with respect to inflation.
Therefore, the distribution theory approach has the merit of
allowing the setting up of smaller, but accurate, reserves.
It is not possible to recomr-iend any one distribution as the
best model for the distribution of general. insurance claim amounts
since different models may fit data from different classes of business.
275
Care must be taken to adopt a model which represents the special
features of the data. For example, if the actual samples consistently
exhibit a mode near the origin, we must use a model like the
lognormal or the inverse Gaussian rather than the Weibull or the
gamma which have no'mode when the value of their shape parameter
is less than 1. If the histograms of the samples have very long
tails then the lognormal, or the inverse Gaussian, would be more
appropriate than the Weibull or the gamma since the former models
have a longer tail. If we are interested in the distribution of
large claims, the truncated lognormal or the Pareto distributions
should be found more suitable. If two models are found to fit
the same data equally well, then the one which has a thicker tail,
and which may overpredict the total cost of claims slightly more,
should be preferred. This is because it is wiser to overpredict
slightly rather than underpredict the total cost of claims. We
recommend the 3-parameter lognormal distribution as the model for
the distribution of the AD claim amounts. The location parameter,
c, of this model was interpreted as the amount of voluntary excess
on the policy. This distribution clearly exhibited the mode and
the long tail of the histograms of the samples. The agreements
between the model and the sample values and between the predicted
models and the sample values were satisfactory. Our proposed
3-parameter inverse Gaussian distribution also presented the above
features and gave almost as good a fit to the data. However, the
lognormal distribution seems more desirable because we have a
theoretical justification for it and because tests of lognormality
are available. In addition a slightly better fit to the ýT data
was provided by this distribution. For the distribution of ]arge
AD claim amounts we recommend the truncated lognormal on the basis
276
of its good fit to the data, but almost as good results may be
obtained by the simpler to use Pareto distribution.
We believe that with the wide availability of computers all
models are relatively easy to use. Therefore, it must be emphasised
that the particular features of the data rather than the ease of use
should be the criteria. in selecting a model.
277
APPFADI K
Computer Programs
pram Description pie
P1 Prints sample frequency and cumulative distribution 220 functions and a set of relevant sample statistics, (section 1.6)1.
P2 Generates, a sample of 2500 lognormal random 281 variates (u = 4.5, a2 = 1) and grcups them according to the format of AD samples. The sample frequency and cumulative distribution functions are printed; (section 3.6.4).
P3 Generates 100 samples of 2500 lognormal random 281 variates (u = 4.5, a2 = 1), (section 3.7.8).
p4 Measures the performance of various methods of 282 estimation on simulated lognomal samples, (section 3.7.8).
P; Estimates parametimrs p and 2, of the 2-parameter 284 e- lognormal distribution, by various methods of estimation, (section 3.8).
P6 Estimates parameters i and a2 , of the 2-parameter 286 i-- lognormal distribution by the AGIL method, (section 3.8).
P7 Plots the histogram of a given sample of AD data 287 and the frequency curve of 2-parameter logno renal model fitted to the actual data (section 3.8).
Performs the prediction of the future distribution 289 of claim amounts: - the 2-parameter lognonal model, (section 3.9.3).
P9 Plots the points (log(x+c), z) for various values 290 of c for the purpose of testing a given sample of data for 3-parameter lognorirality, (section 3.11.2).
,
Pio Estimates the parameters of the 3-parameter 291 logior,,. al distribution by the method of least squares, (section 3.13).
Pll plots the points (log (x+c) , z) , from the sample, 293 and the ]east squares line fitted to them , (section 3.13).
1- For each progr. ^*: the section r. tn: iber refers to the reievasrt section. in the text where the use of that parlAculaz combater prop. =1 i, mentioned for the first time.
? 73
profirar; Description Page
P12 Estimates th': parameters of the 3-parameter 294 lognormal distribution by the M4L method, (section 3.13).
P13 Performs the Weibull graphical test, 295 (section 4.5).
P14 Estimates the parameters of the 3-parameter 296 Weibull distribution by the method of least squares, (section 4.7).
P15 Plots the points (log (x-c) , log log 11x) 297 from the sample, and the least squares line fitted to them, (section 4.7).
P16 Estimates the parameters of the 2-parameter 298 Weibull distribution by the method of wL, (section 4.7).
P17 Estimates the parameters of the 3-parameter 299
. Weibull distribution by the IM method, (section 4.7).
P18 Estimates the parameters of the 2-parameter 301 inverse Gaussian distribution by the b\IL method, (section 5.5).
P19 Estimates the parameters of the 3-parameter 302 inverse Gaussian distribution by the M& method, (section 5.5).
P20 Plots the histogram of a given sample of AD 303 data and the frequency curve of the 3-parameter inverse Gaussian model fitted to the actual data (section S. 5).
P21 Performs the prediction of the future distribution 305 of claim amounts: - the 3-parameter inverse Gaussian model, (section 5.6.2).
P22 Performs the Pareto graphical test, 306 (section 6.5)
P23 Estimates the parameters of the truncated lognormal 307 distribution by the rDM method, (section 7.4)
P24 Estimates the parameter A, of the Pareto 309 distribution of the first kind, by the IMNI, method, (section 7.4).
P25 Estimates the parameters of the Gamii a model by 310 the TIC L method, (section 8.4).
P26 Perfowsthe prediction of the future distribution 312 of claim amounts :- the �a11"la i odel , (section W11.6).
279
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