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Weibull-Pareto Distribution and Its ApplicationsAyman Alzaatreh a , Felix Famoye b & Carl Lee ba Department of Mathematics, Austin Peay State University, Clarksville, Tennessee, USAb Department of Mathematics, Central Michigan University, Mount Pleasant, Michigan, USAVersion of record first published: 01 Apr 2013.

To cite this article: Ayman Alzaatreh , Felix Famoye & Carl Lee (2013): Weibull-Pareto Distribution and Its Applications,Communications in Statistics - Theory and Methods, 42:9, 1673-1691

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Communications in Statistics—Theory and Methods, 42: 1673–1691, 2013Copyright © Taylor & Francis Group, LLCISSN: 0361-0926 print/1532-415X onlineDOI: 10.1080/03610926.2011.599002

Weibull-Pareto Distribution and Its Applications

AYMAN ALZAATREH1, FELIX FAMOYE2,AND CARL LEE2

1Department of Mathematics, Austin Peay State University,Clarksville, Tennessee, USA2Department of Mathematics, Central Michigan University,Mount Pleasant, Michigan, USA

In this article, a new distribution, namely, Weibull-Pareto distribution is definedand studied. Various properties of the Weibull-Pareto distribution are obtained.The distribution is found to be unimodal and the shape of the distribution can beskewed to the right or skewed to the left. Results for moments, limiting behavior,and Shannon’s entropy are provided. The method of modified maximum likelihoodestimation is proposed for estimating the model parameters. Several real data setsare used to illustrate the applications of Weibull-Pareto distribution.

Keywords Modified maximum likelihood estimation; Simulation study; T -Xfamily; Unimodality.

Mathematics Subject Classification 62E15; 62F10; 62P10; 68U20.

1. Introduction

The Weibull distribution is a well-known distribution due to its wide use to modelvarious types of data. Beside the many applications of the Weibull distribution,the distribution has been widely used in survival and reliability analyses.To add more flexibility to Weibull distribution, many researchers developed manygeneralizations of the distribution. These generalizations include the generalizedWeibull distribution by Mudholkar and Kollia (1994), the exponentiated-Weibulldistribution by Mudholkar et al. (1995), and the beta-Weibull distribution byFamoye et al. (2005).

Let F�x� be the cumulative distribution function (cdf) of any random variableX and r�t� be the probability density function (pdf) of a random variable T definedon �0���. The cdf of the generalized family of distributions defined by Alzaatrehet al. (in press) is given by

G�x� =∫ −log�1−F�x��

0r�t�dt� (1.1)

Received December 14, 2010; Accepted June 14, 2011Address correspondence to Felix Famoye, Department of Mathematics, Central

Michigan University, Mount Pleasant, MI 48859, USA; E-mail: felix.famoye@cmich.edu

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The family of distributions defined by (1.1) is called “Transformed-Transformer”family (or T -X family) in Alzaatreh et al. (in press). If a random variable T followsthe Weibull distribution with parameters c and �, r�t� = �c/���t/��c−1e−�t/��c , t ≥ 0,the definition in (1.1) leads to the Weibull-X family with the pdf

g�x� = c

�

f�x�

1− F�x�

{−log�1− F�x��

�

}c−1

exp{−(−log�1− F�x��

�

)c}� (1.2)

In this article, we study a member of Weibull-X family where X is the Paretorandom variable. In Sec. 2, the Weibull-Pareto distribution (WPD) is defined.In Sec. 3, we study the properties of WPD, including the limiting behaviorand unimodality. Moments, the moment-generating function, and the existence ofmoments are studied in Sec. 4. Section 5 deals with estimation and simulation ofWPD. Also in this section, modified maximum likelihood estimation is proposedfor estimating the model parameters. Application of the WPD to real data sets isprovided in Sec. 6.

2. The Weibull-Pareto Distribution

If X is a Pareto random variable with the density function f�x� = k�k/xk+1� x > �,then (1.2) reduces to

g�x� = kc

�x

{k

�log(x�

)}c−1

exp{−(k

�log(x�

))c}� x > �� (2.1)

and replacing k/�, by �, the distribution in (2.1) can be written as

g�x� = �c

x

{� log

(x�

)}c−1exp

{−(� log

(x�

))c}� x > �� c� �� � > 0� (2.2)

A random variable X with the pdf g�x� in (2.2) is said to follow the Weibull-Pareto distribution and will be denoted by WPD�c� �� ��. When � = 1, the WPDreduces to the log-Weibull distribution defined by Sayama and Sekine (2004). Whenc = 1, the WPD reduces to the Pareto distribution with parameters � and �. From(2.2), we obtain the cdf of WPD as

G�x� = 1− exp−�� log�x/���c � (2.3)

In the next section, some general properties of WPD will be addressed includingspecial cases, the limiting behavior, and the unimodality.

3. Properties of the Weibull-Pareto Distribution

The following Lemma gives the relation between WPD and Weibull, exponential,and the Type 1 extreme value distributions.

Lemma 3.1 (Transformation).

(a) If a random variable Y follows the Weibull distribution with parameters c and 1/�,then the random variable X = �eY follows WPD�c� �� ��.

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Weibull-Pareto Distribution and It’s Applications 1675

(b) If a random variable Y follows the standard exponential distribution then therandom variable X = � exp�Y 1/c/�� follows WPD�c� �� ��.

(c) If a random variable Y follows the Type 1 extreme value distribution with scaleparameter 1/c, then the random variable X = � exp�e−Y /�� follows WPD�c� �� ��.

Proof. The results follow by using the transformation technique. �

The hazard function associated with WPD is

hg�x� =g�x�

1−G�x�= �c

x

{� log

(x�

)}c−1� x > �� (3.1)

The limiting behaviors of the Weibull-Pareto pdf and the hazard function are givenin the following theorem.

Theorem 3.1. The limit of the Weibull-Pareto density function and the Weibull-Paretohazard function as x → � is 0 and the limit as x → �+ is given by

limx→�+

g�x� = limx→�+

hg�x� =

0� c > 1�

�/�� c = 1�

�� c < 1�

(3.2)

Proof. We first show that limx→� g�x� = limx→� hg�x� = 0. Since g�x� = hg�x��1−G�x��, we only need to show that limx→� hg�x� = 0. If c ≤ 1 and by definition(3.1), we have limx→� hg�x� = 0. If c > 1, there is an integer m such that 1 < c < m.By using L’Hôpital’s rule, we have

limx→�hg�x� = lim

x→�dm−1

dxm−1�c{� log

(x�

)}c−1

= limx→� c�c − 1��c − 2� · · · �c −m+ 1���� log�x/���c−m/x�

Since c < m, limx→� hg�x� = 0.The result in (3.2) follows directly from the definition of (3.1) and g�x� =

hg�x��1−G�x��. �

The following theorem shows that the Weibull-Pareto distribution is unimodal.

Theorem 3.2. The WPD has a unique mode at x = x0. When c ≤ 1, the mode is x0 = �and when c > 1 the mode x0 is the solution of equation k�x� = 0, where

k�x� = − log�x/��− c�� log�x/���c + c − 1� (3.3)

Proof. The derivative with respect to x of Eq. (2.2) is given by

g′�x� = c�2x−2�� log�x/���c−2 exp−�� log�x/���c k�x�� (3.4)

From (3.4) the critical points of g�x� are x = � and x = x0 where k�x0� = 0.Now, for c ≤ 1, it is easy to see from (3.4) that g′�x� < 0, so g�x� is strictlydecreasing. Also, from Theorem 3.1 we have limx→�+ g�x� = �/� when c = 1 and

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limx→�+ g�x�=� when c < 1. Thus, g�x� has a unique mode at x = �. For c > 1 andusing Theorem 3.1, limx→�+ g�x� = 0 implies that x = � cannot be a modal point. Sothe modes of g�x� are the solutions to the equation k�x� = 0. Finally, we need toshow that equation k�x� = 0 has one solution. The derivative of k�x� with respectto x is given by

k′�x� = −{x−1 + �c2x−1�� log�x/���c−1

}�

which implies that k�x� is strictly decreasing when c > 1 and hence, equation k�x�= 0has at most one solution. Using the facts from Theorem 3.1 that limx→�+ g�x� = 0 andlimx→� g�x� = 0, we conclude that g�x� must have a unique mode. �

In Figs. 1–3, various graphs of g�x� and h�x� are provided. These plots indicatethat the WPD has a very long right rail, and when the parameter � increases thepeak of the distribution increases.

Quantile functions for many distributions do not have closed form. For theWeibull-Pareto distribution, the following Lemma gives a closed form quantilefunction of the distribution.

Lemma 3.2. Let Q���� 0 < � < 1 denote the quantile function for the WPD. Then, Q���is given by

Q��� = � exp(�− log�1− ���1/c/�

)� (3.5)

Figure 1. The Weibull-Pareto pdf for various values of c when � = 1 and � = 1.

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Weibull-Pareto Distribution and It’s Applications 1677

Figure 2. The Weibull-Pareto pdf when � = 3 and various values of c and �.

Figure 3. The Weibull-Pareto hazard function for various values of c and � when � = 1and � = 1.

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Proof. By using G�Q���� = � and (2.3), we obtain (3.5), the quantile function ofWPD. �

Setting � = 0�25� 0�50, and 0�75 in (3.5), the quartiles of theWPD can be obtained.The entropy of a random variable X is a measure of variation of uncertainty

(Rényi, 1961). Shannon’s entropy for a random variable X with pdf g�x� is definedas E− log�g�X��. Shannon (1948) showed important applications of this entropyin communication theory. The entropy has also been used in many fields such asphysics, engineering, and economics.

Lemma 3.3. The Shannon’s entropy for a random variable X that follows the WPD is

− log��c/��+ �1+ c−1�/� + �1− c−1��+ 1� (3.6)

where � = − ∫ �0 e−u log udu ≈ 0�57722 is the Euler gamma constant.

Proof. For WPD, the Shannon’s entropy is given by E− log�g�X�� = − ∫ ��g�x�

log�g�x��dx. The result in (3.6) is derived by using the substitution u = �� log�x/���c

in g�x� of (2.2). �

4. Moments

The moment generating function for the WPD is given by

MX�t� = E�etX� =∫ �

�etxg�x�dx =

∫ �

�

( �∑i=0

tixi

i! g�x�

)dx =

�∑i=0

ti

i!E�Xi�� (4.1)

If X follows the WPD, then Lemma 3.1 implies that the random variableY = log�X/�� follows the Weibull distribution with parameters c and 1/� andhence, MY�t� =

∑�i=0

ti �1+i/c�

i!�i . Since X = �eY , we have E�X� = �E�eY � = �MY �1� =�∑�

i=0 �1+i/c�

i!�i . In general, the sth non central moment is given by

E�Xs� = �sE�esY � = �sMY �s� = �s�∑i=0

si

i!�i �1+ i/c�� (4.2)

The result in (4.2) can be used in (4.1) to re-write the moment generating function as

MX�t� =�∑i=0

{�iti

i!�∑j=0

ij �1+ j/c�

j!�j

}� (4.3)

The sth non central moment in (4.2) may not exist for all values of � and c.Theorem 3.3 gives the conditions for the existence of the non central moments.

Theorem 4.1.

(i) If c > 1, then the non central moments of the Weibull-Pareto distribution exist.(ii) If c < 1, then the non central moments of the Weibull-Pareto distribution do not

exist.(iii) If c = 1, then the non central moments of the Weibull-Pareto distribution exist iff

� > s.

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Weibull-Pareto Distribution and It’s Applications 1679

Proof. We need the following inequalities introduced by Keckic and Vasic (1971)for the proof:

�b�

�a�<

bb−1/2

aa−1/2e−�b−a�� 0 < a < b� (4.4)

�b�

�a�>

bb−1

aa−1e−�b−a�� 0 < a < b� (4.5)

(i) When c > 1: From (4.2), E�Xs� = �s∑�

i=0 ai where ai = si �1+i/c�

i!�i . By usingthe series ratio test, we have

limi→�

ai+1

ai

= limi→�

s ��i+ 1�/c��i �i/c�

� (4.6)

On using the inequality in (4.4) and the fact that c > 1, the result in (4.6) becomes

limi→�

ai+1

ai

< limi→�

se−1/c

��1+ 1/i�i/c�1+ 1/i�−1/2 ��i+ 1�/c�1/c

i

= s

�limi→�

��i+ 1�/c�1/c

i= 0�

Hence, E�Xs� exists.

(ii) When c < 1: Using the inequality in (4.5) and the fact that c < 1, the resultin (4.6) becomes

limi→�

ai+1

ai

>s

�limi→�

��i+ 1�/c�1/c

i= ��

Hence, E�Xs� does not exist.

(iii) When c = 1: If c = 1, then

E�Xs� = �s�∑i=0

si

�i�

which exists if and only if � > s. �

The next theorem shows that when � > s and c ≥ 1, the non central momentsof WPD�c� �� �� is bounded above by the non central moments of the Paretodistribution with parameters � and �.

Theorem 4.2. If c ≥ 1 and � > s, then E�Xs� ≤ �s�/�� − s�.

Proof. Since �x� is an increasing function for x ≥ 2 and for any integer i andc ≥ 1, 1+ i/c ≤ 1+ i. Thus, for large i we have �1+ i/c�/ �1+ i� ≤ 1. Hence,

E�Xs� = �s�∑i=0

si �1+ i/c�

�i �1+ i�≤ �s

�∑i=0

si

�i= �s�

� − s� whenever � > s�

�

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Table 1Mode, median, mean, variance, skewness, and kurtosis for some values

of c and � with � = 1 (*: Undefined)

c � Mode Median Mean Variance Skewness Kurtosis

1 1.001 1 1.9986 1001 * * *4 1 1.1892 1.3333 0�2222 7�0711 *7 1 1.1041 1.1667 0�0389 3�3806 22�857010 1 1.0718 1.1111 0�0154 2�8111 17�8290

4 1.001 2�3511 2.4881 2.5535 0�4211 0�5903 3�36924 1�2562 1.2562 1.2569 0�0064 0�0789 2�76267 1�1405 1.1392 1.1390 0�0017 0�0076 2�745510 1�0967 1.0955 1.0952 0�0008 −0�0208 2�7432

7 1.001 2�5945 2.5806 2.5768 0�1540 −0�0235 2�80344 1�2752 1.2678 1.2644 0�0024 −0�3488 3�02527 1�1494 1.1452 1.1433 0�0007 −0�3974 3�088610 1�1025 1.0995 1.0982 0�0003 −0�4170 3�1165

10 1.001 2�6571 2.6197 2.6033 0�0830 −0�2907 2�97424 1�2798 1.2725 1.2690 0�0013 −0�5465 3�37217 1�1516 1.1476 1.1457 0�0003 −0�5851 3�452510 1�1039 1.1012 1.0999 0�0002 −0�6008 3�4865

From Theorem 4.2, the mode of WPD is at � when c ≤ 1. Table 1 providesthe mode, median, mean, variance, skewness, and kurtosis of the WPD for variousvalues of � and c when � = 1. For fixed c > 1 and �, the mode, median, and meanof the WPD are decreasing functions of �. Also, for fixed � > 1 and �, the modeand the median of the WPD are increasing functions of c. When c > 1, the varianceof the WPD is a decreasing function of c and �. The distribution of the WPD tendsto be skewed more to the left as c increases.

5. Parameter Estimation

In this section, we address two problems when using the ML method to estimatethe WPD parameters. The first problem occurs when c < 1. The WPD likelihoodfunction tends to infinity as � approaches the sample minimum x�1� and hence, whenc < 1 and � is estimated by x�1�, no MLE for c and � exists. This problem wasstudied by Smith (1985) who proposed an alternative approach for estimating theparameters. The second problem is when c � 1. For this situation, the WPD has along left tail which makes x�1� a poor estimate for � and this produces an unusuallylarge bias in the alternative MLE for c and �. A modification of the regular MLEis proposed to deal with this large bias problem. Simulation study is conducted toevaluate the performance of the alternative and the modified MLE methods.

5.1. Alternative Maximum Likelihood Estimation

Smith (1985) considered probability densities of the form

f�x� �� �� = �x − ��c−1q�x − �� ��� for x > �� (5.1)

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Weibull-Pareto Distribution and It’s Applications 1681

where parameter � and parameter vector � are unknown. The WPD in (2.2) is ofthe form (5.1) with � = ��� c�. The classical regularity conditions for the maximumlikelihood estimates are not satisfied for the WPD because the support of the densitydepends on �. Smith (1985) showed that the classical MLE results hold for (5.1)when c > 2 and studied in detail the case when c ≤ 2. For any value of c, Smith(1985) proposed the following alternative to maximum likelihood. If a randomsample x1� x2� � � � � xn is obtained, estimate the parameter � by the sample minimumx�1� and then use the MLE method to estimate � and c by excluding the sampleminimum. We apply Smith’s alternative MLE (AMLE) approach to estimate theparameters of WPD and conduct a simulation study to evaluate the performance ofthe AMLE for WPD.

The alternative log-likelihood function for WPD is given by

L∗ =∑

xi =x�1�

log g�xi� x�1�� �� c�

= ∑xi =x�1�

{c log � + log c − log xi + �c − 1� log�log�xi/x�1���

− �� log�xi/x�1���c}� (5.2)

The derivatives of (5.2) with respect to � and c are given by

�L∗��

= ∑xi =x�1�

{c/� − c�c−1

(log�xi/x�1��

)c}(5.3)

�L∗�c

= ∑xi =x�1�

{c + log � + log

(log(xi/x�1�

))− log �(� log

(xi/x�1�

))c− (

� log(xi/x�1�

))clog(log(xi/x�1�

)) }� (5.4)

Setting (5.3) and (5.4) to zero and simplifying we obtain

� ={�n− n′�/

∑xi =x�1�

(log�xi/x�1��

)c}1/c

� (5.5)

c−1 + ∑xi =x�1�

log

(log

(xix�1�

))−∑

xi =x�1�

(log�xi/x�1��

)clog(log�xi/x�1��

)∑

xi =x�1�

(log�xi/x�1��

)c = 0� (5.6)

where n′ is the frequency of x�1�. The AMLE c of c is the solution of Eq. (5.6). TheAMLE � of � can be found by substituting the estimate c in Eq. (5.5).

5.2. Simulation Study to Evaluate the Performance of the AMLE

Smith (1985) showed the AMLE is a consistent estimator. However, it is not clearhow the estimator performs in terms of bias for small samples. We conduct asimulation study to evaluate the AMLE of WPD in terms of both bias and variancefor various parameter combinations and different sample sizes. We consider thevalues 0�5� 1, and 3 for parameters � and �, and 0�5� 1� 4, and 7 for the parameterc. Two different sample sizes n = 100 and 500, are considered. The simulation is

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done for a total of 36 parameter combinations. For each parameter combination, wegenerate a random sample y1� y2� � � � � yn from Weibull distribution with parametersc and 1/�. By using the transformation xi = � exp�yi� (see Lemma 3.1), we obtain arandom sample x1� x2� � � � � xn, which follow WPD�c� �� ��.

The initial values for the parameters � and c (Johnson et al., 1994, pp. 642–643)are c0 = �/�

√6slog�yi�� and �0 = exp�−xlog�yi� − �/c0�, where slog�yi� and xlog�yi� are the

sample standard deviation and the sample mean for log�yi� and � is the Eulergamma constant. This process is repeated 200 times. The bias (estimate − actual)and the standard deviation are presented in Tables 2 and 3.

The result of the simulation shows that the AMLE method does not providegood estimates when c > 1. When c > 1, the estimates of c and � are far from theactual values, because the estimates of c and � are very sensitive to the estimateof �. If � = x�1� is greater than � by a small quantity, it gives a large bias forestimating c using the AMLE. This bias comes from the fact that c is the solutionof Eq. (5.6). If � is greater than � by a small quantity, then the term log�log�xi/���becomes very negatively large when xi is close to � and hence the value of c becomessmaller than the actual value. For example, if � = 1, � = 1�3, and xi = 1�3001, thenlog�log�xi/��� = −9�4727, while the actual value log�log�xi/��� = −1�3377. FromTables 2 and 3, it appears that � = x�1� overestimates � and becomes worse as cincreases. A closer look at the shape of the WPD (see Figs. 1, 2, and Table 2), itindicates that the WPD is gradually shifting from right-skewed to left-skewed asc increases. The sample minimum, x�1�, tends to be larger than the lower bound �of the distribution when the distribution shape is more left-skewed and the samplesize is small. This explains why x�1� in small samples overestimates � more seriouslyas c increases in WPD. In the next sub-section, a modified maximum likelihoodestimation (MMLE) is proposed and the results show better estimates (in terms ofthe biases) for the parameters c, �, and �.

5.3. Modified Maximum Likelihood Estimation

As observed and explained in Sec. 5.2, the large bias of AMLE is a problem,especially when the shape of WPD is more left-skewed (when the parameter c islarge) and the sample size is small. In view of this problem, there is a need to seeka different approach to estimate �. To address this problem, we consider a modifiedMLE proposed by (Smith, 1985, Sec. 3) for the density function of the form (5.1).Smith showed that the parameter estimates exist and are consistent when c > 1. Weconsider the log-likelihood function

Ln�c� �� �� =n∑

i=1

log g�xi� �� �� c�� (5.7)

which is defined only for � < x�1�.The estimators c� �, and � satisfy the following equations:

�Ln�c� �� ��

�c= 0�

�Ln�c� �� ��

��= 0� and

�Ln�c� �� ��

��= 0� (5.8)

In order to use (5.8), we need to show that the derivative with respect to � existswhenever � < x�1� for the WPD. The derivative of Ln�c� �� �� with respect to � for

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Weibull-Pareto Distribution and It’s Applications 1683

Table 2Bias and standard deviation of the parameter estimates

using AMLE method �n = 100�

Actual values Bias Standard deviation

c � � c � � c � �

0.5 0�5 0�5 0�0153 0�0194 0.0002 0.0396 0.1098 0.00061 0�0164 0�0136 0.0003 0.0396 0.1129 0.00093 0�0186 −0�0064 0.0012 0.0401 0.1032 0.0027

1 0�5 0�0215 0�0123 0.0001 0.0418 0.2241 0.00021 0�0203 0�0091 0.0002 0.0421 0.2002 0.00043 0�0199 −0�0108 0.0007 0.0397 0.2032 0.0014

3 0�5 0�0266 −0�0452 0.0000 0.0413 0.6353 0.00011 0�0294 −0�0086 0.0001 0.0409 0.6388 0.00013 0�0274 −0�0246 0.0003 0.0410 0.6741 0.0006

1 0�5 0�5 0�0170 0�0032 0.0098 0.0871 0.0501 0.00961 0�0276 0�0160 0.0226 0.0835 0.0544 0.02193 0�0105 0�0154 0.0649 0.0815 0.0605 0.0757

1 0�5 0�0101 0�0213 0.0052 0.0805 0.0985 0.00541 0�0161 0�0227 0.0096 0.0783 0.1075 0.01013 0�0196 0�0188 0.0324 0.0792 0.1092 0.0322

3 0�5 0�0180 0�0898 0.0019 0.0798 0.3789 0.00201 0�0220 0�0589 0.0035 0.0734 0.3382 0.00333 0�0147 0�0399 0.0088 0.0739 0.3530 0.0086

4 0�5 0�5 −1�2733 0�2339 0.4105 0.4378 0.1020 0.15471 −1�3236 0�2371 0.8291 0.3673 0.0902 0.27523 −1�2307 0�2346 2.4486 0.4472 0.0946 0.8827

1 0�5 −1�2482 0�4718 0.1722 0.4435 0.1840 0.05321 −1�1904 0�4429 0.3274 0.4121 0.1698 0.10203 −1�2342 0�4624 1.0173 0.4361 0.2007 0.3418

3 0�5 −1�1801 1�3751 0.0508 0.4370 0.5605 0.01451 −1�1713 1�3655 0.1017 0.4481 0.5534 0.03053 −1�1542 1�3481 0.3008 0.4369 0.5488 0.0894

7 0�5 0�5 −3�5316 0�5030 0.8145 0.6846 0.1708 0.21501 −3�6250 0�5242 1.6773 0.6764 0.1726 0.41953 −3�5337 0�5107 4.9343 0.6644 0.1711 1.2583

1 0�5 −3�5971 1�0701 0.3188 0.6914 0.3616 0.06761 −3�5810 1�0515 0.6342 0.6608 0.3383 0.13313 −3�5575 1�0535 1.9036 0.6525 0.3372 0.3831

3 0�5 −3�4027 3�0641 0.0874 0.7356 1.0182 0.01601 −3�4350 3�1598 0.1775 0.7834 1.0427 0.03293 −3�4386 3�0456 0.5220 0.7045 1.0407 0.0964

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Table 3Bias and standard deviation of the parameter estimates

using AMLE method �n = 500�

Actual values Bias Standard deviation

c � � c � � c � �

0.5 0�5 0�5 0�0026 0�0037 0.0000 0.0171 0.0511 0.00001 0�0076 −0�0083 0.0000 0.0172 0.0451 0.00003 0�0058 0�0054 0.0000 0.0170 0.0474 0.0001

1 0�5 0�0030 −0�0007 0.0000 0.0153 0.1000 0.00001 0�0077 0�0110 0.0000 0.0179 0.0968 0.00003 0�0052 −0�0020 0.0000 0.0170 0.0922 0.0001

3 0�5 0�0056 0�0011 0.0000 0.0185 0.2797 0.00001 0�0065 −0�0080 0.0000 0.0185 0.2892 0.00003 0�0053 0�0239 0.0000 0.0154 0.3073 0.0000

1 0�5 0�5 0�0032 0�0034 0.0019 0.0323 0.0212 0.00201 0�0017 −0�0008 0.0042 0.0324 0.0222 0.00463 −0�0013 0�0036 0.0145 0.0362 0.0240 0.0137

1 0�5 0�0031 0�0029 0.0010 0.0339 0.0481 0.00091 0�0041 0�0018 0.0020 0.0346 0.0501 0.00193 0�0030 0�0062 0.0062 0.0334 0.0512 0.0068

3 0�5 0�0034 0�0002 0.0003 0.0351 0.1361 0.00031 0�0024 0�0047 0.0006 0.0345 0.1378 0.00053 0�0046 −0�0002 0.0022 0.0328 0.1401 0.0021

4 0�5 0�5 −0�8491 0�1241 0.2295 0.2706 0.0437 0.07701 −0�8710 0�1288 0.4704 0.2604 0.0463 0.16273 −0�8605 0�1259 1.3939 0.2742 0.0459 0.4703

1 0�5 −0�9010 0�2687 0.1109 0.2601 0.0890 0.03161 −0�8693 0�2618 0.2162 0.2703 0.0859 0.06203 −0�8813 0�2628 0.6488 0.2965 0.0976 0.2077

3 0�5 −0�8547 0�7623 0.0326 0.2763 0.2669 0.00941 −0�8514 0�7623 0.0661 0.2746 0.2760 0.01903 −0�8892 0�7939 0.2027 0.2576 0.2548 0.0539

7 0�5 0�5 −2�9279 0�3349 0.5892 0.5038 0.0921 0.13901 −2�8761 0�3243 1.1449 0.5421 0.0939 0.27983 −2�9323 0�3365 3.5438 0.5535 0.0962 0.8767

1 0�5 −2�9542 0�6894 0.2417 0.5098 0.1834 0.04711 −2�9473 0�6812 0.4786 0.5106 0.1827 0.09253 −2�9500 0�6870 1.4438 0.5396 0.1851 0.2882

3 0�5 −2�9133 2�0201 0.0690 0.5207 0.5345 0.01191 −2�9147 2�0196 0.1382 0.4955 0.5185 0.02283 −2�8888 2�0153 0.4135 0.5313 0.5474 0.0729

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Weibull-Pareto Distribution and It’s Applications 1685

the WPD is

�Ln�c� �� ��

��= 1

�

n∑i=1

1− c

log�xi/��+ c�c

�

n∑i=1

�log�xi/���c−1� (5.9)

It is clear that the right-hand side of Eq. (5.9) is continuous when 0 < � < x�1� andhence, �Ln�c�����

��exists. Setting Eq. (5.9) to zero, we get

n∑i=1

1− c

log�xi/��+ c� c

n∑i=1

�log�xi/���c−1 = 0� (5.10)

and setting �Ln�c�����

��= 0 and �Ln�c�����

�c= 0, we get

� ={n

/n∑

i=1

�log�xi/���c

}1/c

� (5.11)

c−1 +n∑

i=1

log �log�xi/���−∑n

i=1 �log�xi/���c log �log�xi/���∑n

i=1 �log�xi/���c = 0� (5.12)

The estimators c� �, and � from (5.10)–(5.12) will be called MMLE.

Table 4Bias and standard deviation of the parameter estimates

using MMLE method �n = 100�

Actual values Bias Standard deviation

c � � c � � c � �

4 0�5 0�5 −0�1791 0.0505 0.0877 0.9675 0.1153 0.20281 −0�1059 0.0492 0.1692 0.9713 0.1117 0.40753 −0�1432 0.0498 0.5138 0.9646 0.1088 1.1955

1 0�5 0�1311 0.0568 0.0098 1.3075 0.2551 0.11461 −0�0680 0.0904 0.0538 1.0991 0.2444 0.21173 −0�1264 0.0948 0.1811 0.9929 0.2179 0.5863

3 0�5 −0�1407 0.2693 0.0078 0.9301 0.6689 0.03201 −0�1644 0.2781 0.0161 0.9441 0.6790 0.06323 −0�0039 0.2065 0.0263 1.0276 0.6606 0.1981

7 0�5 0�5 −1�2633 0.1524 0.2709 1.5586 0.1793 0.30441 −1�4256 0.1660 0.5902 1.4926 0.1735 0.59673 −1�2287 0.1595 1.6752 1.7056 0.2009 1.9508

1 0�5 −0�4250 0.2123 0.0616 2.7144 0.4160 0.16861 −1�0953 0.3312 0.2193 2.1728 0.4198 0.29423 −1�1783 0.3351 0.6818 2.3534 0.4085 0.9044

3 0�5 −1�8528 1.2261 0.0436 1.2541 0.9598 0.03031 −1�4892 1.0231 0.0721 1.4055 1.0194 0.06903 −0�4569 0.6521 0.0849 2.5257 1.2771 0.3385

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Table 5Bias and standard deviation of the parameter estimates

using MMLE method �n = 500�

Actual values Bias Standard deviation

c � � c � � c � �

4 0�5 0�5 0�0053 0.0064 0.0119 0.3979 0.0424 0.07971 −0�0016 0.0063 0.0241 0.4676 0.0508 0.19473 −0�0143 0.0050 0.0555 0.4024 0.0434 0.4870

1 0�5 −0�0285 0.0169 0.0052 0.4173 0.0913 0.04281 −0�0156 0.0141 0.0079 0.4039 0.0933 0.08633 −0�0612 0.0241 0.0543 0.4187 0.0952 0.2633

3 0�5 −0�0551 0.0783 0.0029 0.4320 0.2732 0.01461 −0�1084 0.1017 0.0079 0.3691 0.2578 0.02633 −0�0242 0.0437 0.0046 0.4628 0.3023 0.0942

7 0�5 0�5 −0�2499 0.0315 0.0597 1.0957 0.0836 0.15941 −0�2099 0.0265 0.1020 1.0401 0.0742 0.28713 −0�4188 0.0431 0.5019 0.9509 0.0767 0.8855

1 0�5 −0�2271 0.0607 0.0214 1.2592 0.1780 0.08221 −0�0999 0.0543 0.0352 1.5578 0.1845 0.18153 −0�5093 0.0952 0.2354 0.9675 0.1563 0.4127

3 0�5 −0�7587 0.3871 0.0172 0.7900 0.4244 0.01881 −0�7230 0.3926 0.0338 0.8624 0.4798 0.04023 −0�2309 0.1880 0.0383 1.2144 0.5059 0.1564

Applying the same simulation study as in Sec. 5.2, we obtain the results thatare summarized in Tables 4 and 5. Since the MMLE are proposed for c > 1 andthe MMLE exist and are consistent when c > 1, the simulation is done for valuesof c > 1.

The results in Tables 4 and 5 show an improvement in estimating the parameter� in terms of bias, which in turn provides improved parameter estimates for c and �in terms of bias. However, the standard deviation of the MMLE is higher than thoseof the AMLE. To compare the performance of AMLE with MMLE when c > 1,we compute the mean square error (MSE) of the simulation results for AMLE andMMLE when c > 1. In order to conserve space, these MSE are not reported. Theresults indicate that MMLE consistently has smaller MSE than AMLE when c > 1.In practice, one should first obtain a graphical display of the data to be fitted.If the data has a reversed J-shape, we recommend the use of AMLE method forestimation. Otherwise, we suggest using the MMLE method since biases are reduceddramatically when compared to AMLE. Further research is needed for developingbetter estimation method as well as obtaining the asymptotic distribution of theparameter estimators.

6. Application

In this section, the WPD is applied to model three data sets from Park et al. (1964)and Park (1954). The three data sets represent the grouped frequency distributions

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Weibull-Pareto Distribution and It’s Applications 1687

of adult numbers for Tribolium Confusum and Tribolium Castaneum culturedat 24�C and Tribolium Confusum strain. Tables 6–8 summarize the results offitting these data to the generalized Weibull distribution (Mudholkar et al., 1996),exponentiated-Weibull distribution (Mudholkar et al., 1995), Lagrange-gammadistribution (Famoye and Govindarajulu, 1998), and Weibull-Pareto distribution.Famoye and Govindarajulu (1998) used the method of moments to estimate theLagrange-gamma parameters. Method of maximum likelihood estimation was used

Table 6Observed and expected frequencies for Tribolium Castaneum cultured at 24�C

Expected

x-value ObservedGeneralizedWeibull

Lagrange-gamma

Exponentiated-Weibull

Weibull-Pareto

20–30 2 6.86 3.31 4.45 3.3030–40 15 14.19 10.06 12.06 12.0940–50 26 24.04 22.36 23.68 25.1650–60 30 35.80 38.06 37.86 39.8760–70 67 48.37 54.09 52.34 53.8170–80 67 60.32 67.54 64.86 65.1880–90 65 70.13 76.50 73.74 72.9190–100 80 76.49 80.32 78.14 76.59100–110 72 78.64 79.35 78.10 76.38110–120 70 76.52 74.58 74.30 72.86120–130 77 70.78 67.24 67.73 66.85130–140 59 62.51 58.53 59.52 59.24140–150 47 52.95 49.44 50.63 50.88150–160 39 43.25 40.68 41.87 42.45160–170 29 34.24 32.71 33.76 34.49170–180 25 26.39 25.78 26.61 27.34180–190 24 19.90 19.96 20.55 21.18190–200 19 14.75 15.21 15.58 16.05200–210 19 10.78 11.42 11.61 11.92210–220 7 7.80 8.46 8.51 8.69220–230 6 5.60 6.20 6.15 6.21230–240 4 4.01 4.49 4.39 4.37240–250 3 2.86 3.22 3.08 3.02250–260 4 2.04 2.35 2.14 2.05260–270 1 7.78 5.14 5.33 4.09

}Total 857 857 857 857 857

Parameter estimates � = 0�3093 r = 7 � = 1�6777 c = 6�9525� = 120�83 � = 0�0262 � = 87�0155 � = 9�1065� = −0�2963 � = 0�24878 � = 2�7099 � = 0�3802

�2 27.69 21.02 19.57 17.23df 20 20 20 20p-value 0.1169 0.3957 0.4853 0.6380

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1688 Alzaatreh et al.

to estimate the generalized Weibull and the exponentiated-Weibull parameters. Themodified maximum likelihood estimation method (Sec. 5.3) is used to estimate theWeibull-Pareto parameters.

The results in Table 6 indicate that the Weibull-Pareto distribution providesthe best fit among the four distributions, while the exponentiated-Weibull andthe Lagrange-gamma also provide adequate fit to the data. In examining thedistribution of this data, we notice that the data has a very long right tail.

Table 7Observed and expected frequencies for Tribolium Confusum cultured at 24�C

Expected

x-value ObservedGeneralizedWeibull

Lagrange-gamma

Exponentiated-Weibull

Weibull-Pareto

20–30 0 0.47 0.02 0.05 0.0030–40 0 2.01 0.41 0.57 0.0040–50 3 5.96 2.91 3.33 2.37

50–60 9 14.15 11.38 11.81 12.8260–70 39 28.48 29.57 29.30 32.9770–80 53 49.96 57.00 55.45 59.0480–90 77 77.04 87.64 84.72 85.1190–100 105 104.29 112.87 108.87 105.46100–110 135 123.36 126.05 121.41 116.18110–120 114 127.42 125.27 120.42 116.08120–130 113 115.77 112.93 108.36 106.54130–140 92 94.02 93.75 89.90 90.64140–150 59 69.79 72.52 69.67 71.97150–160 54 48.48 52.77 50.98 53.61160–170 38 32.22 36.41 35.52 37.63170–180 22 20.86 23.97 23.71 24.99180–190 17 13.32 15.13 15.25 15.75190–200 6 8.48 9.21 9.48 9.44200–210 10 5.42 5.42 5.72 5.40210–220 3 3.48 3.10 3.35 2.96220-230 2 2.26 1.72 1.91 1.55230–240 0 1.49 0.93 1.06 0.78240–250 1 0.99 0.50 0.57 0.38250–260 0 0.66 0.26 0.30 0.18260–270 0 1.61 0.26 0.30 0.14

Total 952 952 952 952 952

Parameter estimates � = 0�1838 r = 14 � = 1�8776 c = 5�0587� = 118�86 � = 0�06502 � = 77�4141 � = 33�2082

� = −0�5831 � = 0�06582 � = 5�3097 � = 0�7473

�2 23.02 17.22 14.26 15.26df 14 14 14 14p-value 0.0599 0.2448 0.4303 0.3605

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Weibull-Pareto Distribution and It’s Applications 1689

This example suggests that the Weibull-Pareto distribution performs very well incapturing a long right tail characteristic.

The results in Table 7 indicate that the exponentiated-Weibull distribution fitthe data set the best, the Weibull-Pareto is a close second, then, followed by theLagrange-gamma. The generalized Weibull does not provide a good fit comparedto the other distributions. The distribution of this data shows that the data has avery long right tail and a noticeable left tail as well. This example suggests that theWeibull-Pareto distribution does very well in fitting the distributions of data withunusually long left and right tails characteristic.

The results in Table 8 indicate that the Weibull-Pareto distribution fits the bestamong the four distributions, while the exponentiated-Weibull and the Lagrange-gamma also provide adequate fit to the data. The distribution of this data showsthat the data is approximately symmetric. This example suggests that the Weibull-Pareto distribution is capable of fitting very well the distribution of data whichis somewhat symmetric. Among all the four distributions, only the Weibull-Paretodistribution provides the best fit to both the right and left tails.

Table 8Observed and expected frequencies for Tribolium Confusum strain

Expected

x-value ObservedGeneralizedWeibull

Lagrange-gamma

Exponentiated-Weibull

Weibull-Pareto

35–40 5 4.15 2.29 3.06 2.8340–45 5 8.21 6.33 7.53 7.98

}45–50 14 14.74 15.30 15.33 16.3850–55 33 24.02 28.25 26.30 27.2455–60 40 35.36 41.83 38.48 38.5460–65 49 46.41 51.47 48.53 47.5765–70 44 53.48 54.12 53.16 51.8470–75 52 53.33 49.73 50.98 50.1575–80 44 45.63 40.65 43.10 43.1480–85 28 33.58 30.00 32.35 33.0085–90 29 21.53 20.24 21.71 22.4290–95 13 12.30 12.61 13.08 13.5195–100 9 6.42 7.31 7.11 7.21100–105 1 3.15 3.98 3.49 3.41105–110 1 1.48 2.05 1.55 1.42110–115 1 4.19 1.84 2.24 1.36

Total 368 368 368 368 368

Parameter estimates � = 0�1498 r = 16 � = 3�1559 c = 6�1612� = 72�1687 � = 0�16260 � = 58�9881 � = 23�2825

� = −0�3299 � = 0�05351 � = 3�2394 � = 0�8657

�2 14.80 10.94 9.60 7.78df 9 9 9 9p-value 0.0966 0.2797 0.3842 0.5562

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1690 Alzaatreh et al.

In summary, the results from these three examples indicate that the Weibull-Pareto distribution provides very adequate fits to these three types of distributions(very long right tail, left tail, and somewhat symmetric). The distributions of WPDdisplayed in Figs. 1 and 2 indeed show that the distribution is very flexible and canfit very well a wide range of data sets.

7. Conclusion

A special case of the Weibull-X family, the Weibull-Pareto distribution, isdefined and studied. Various properties of the Weibull-Pareto distribution arestudied, including the moments, moment generating function, hazard function, andunimodality. Ordinary maximum likelihood estimation method cannot be usedto estimate the WPD parameters when c < 1. Two methods of estimation areproposed, AMLE and MMLE. AMLE appears to give poor parameter estimates forWPD when c is large. The results of a simulation study in Tables 2–5 show that theMMLE produces better estimates than the AMLE. Three real data sets are fittedto the WPD and compared with other known distributions. The results show thatthe WPD gives a good fit to each data set and provides the best fit to the right andleft tails. The Weibull-Pareto distribution can be a good model to fit data with longright tail as well as long left tail.

Acknowledgment

The authors are grateful for the comments and suggestions by the referee and theEditor. Their comments and suggestions greatly improved the article.

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