ARTICLE IN PRESS

Statistics & Probability Letters 72 (2005) 51–57

0167-7152/$ -

doi:10.1016/j.

�CorresponE-mail add

ahmadi@mat

www.elsevier.com/locate/stapro

Characterization of hazard function factorization by Fisherinformation in minima and upper record values

Glenn Hofmanna,�, N. Balakrishnanb, Jafar Ahmadic

a5313 N. Ravenswood, Unit 201, Chicago, IL 60640, USAbDepartment of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ont.,

Canada L8S 4K1cSchool of Mathematical Sciences, Ferdowsi University of Mashhad, P.O. Box 91775-1159, Mashhad, Iran

Received 23 June 2004; received in revised form 6 December 2004; accepted 20 December 2004

Available online 26 January 2005

Abstract

The hazard function is an important characteristic for the analysis of reliability data. It is therefore ofinterest to see under what conditions it can be expressed as the product of a function of the variable and afunction of the parameter. We show that such a factorization can be characterized by the property of Fisherinformation in minima and upper record values. We present similar results for the reversed hazard rate bythe property of Fisher information in maxima and lower record values. These properties imply thecharacterization of two classes of exponential families.r 2005 Elsevier B.V. All rights reserved.

Keywords: Fisher information; Order statistics; Record values; Characterization; Hazard function; Exponential family;

Reversed hazard rate; Minimal repairs

1. Introduction

Let X be a random variable with cumulative distribution function (cdf) Fðx; yÞ and probabilitydensity function (pdf) f ðx; yÞ; where y is a scalar parameter with y 2 Y; Y being an open set.

see front matter r 2005 Elsevier B.V. All rights reserved.

spl.2004.12.002

ding author.

resses: glennhofmann@yahoo.com (G. Hofmann), bala@univmail.cis.mcmaster.ca (N. Balakrishnan),

h.um.ac.ir (J. Ahmadi).

ARTICLE IN PRESS

G. Hofmann et al. / Statistics & Probability Letters 72 (2005) 51–5752

Denote the hazard function of X by hðx; yÞ ¼ f ðx; yÞ=Fðx; yÞ; where F ðx; yÞ ¼ 1� Fðx; yÞ: Undercertain regularity conditions (see, for example, Lehmann and Casella, 1998, p. 115 or Rao, 1973,p. 329) the Fisher information about y that is contained in X, is defined by

IðX ; yÞ ¼ Eqqylog f ðX ; yÞ

� �2" #.

In this paper, we assume these regularity conditions to hold. Another expression for IðX ; yÞ can bederived from the work of Efron and Johnstone (1990), who studied the identity

IðX ; yÞ ¼ Eqqylog hðX ; yÞ

� �2" #. (1)

Let X i:n be the ith order statistic from the random sample X 1; . . . ;X n from F ðx; yÞ:The problem of obtaining the Fisher information in order statistics was described by Arnold et

al. (1992) with the words: ‘‘While the recipe for IY ðyÞ is simple, the details are messy in mostcases’’, where Y is an arbitrary collection of order statistics. Several results have been published inthis direction in recent years. We will briefly state a few examples. Mehrotra et al. (1979) presentedthe Fisher information in the first r order statistics. Park (1996) used an indirect approach toobtain the Fisher information in the rth order statistic, and presented very informative plots todemonstrate what order statistics have more Fisher information. Zheng and Gastwirth (2000)calculated the Fisher information contained in any collection of order statistics. Abo-Eleneen andNagaraja (2002) studied the Fisher information in collections of order statistics and theircontaminants from bivariate samples. In the exponential family case, Iyengar et al. (1999) gave ageneral condition on F such that X k:n has more Fisher information than a single observation.Zheng and Gastwirth (2001) recently discussed the loss in Fisher information under randomcensoring.Zheng (2001) presented a characterization of the hazard function which is a product of a

function of the variable and a function of the parameter using the Fisher information in the first rorder statistics. The extreme value distribution (location parameter) and the Weibull distribution(scale parameter) are important examples of this type. Specifically, he proved that ifIðX 1:n; . . . ;X r:n; yÞ ¼ rIðX ; yÞ holds for all nX1 and all 1prpn; then the hazard function hasthe form hðx; yÞ ¼ cðxÞZðyÞ; where cðxÞ and ZðyÞ are some positive functions, of x and y alone,respectively. In Section 2, we show that the condition for this characterization can be weakened. Itis sufficient if it holds for r ¼ 1 and all n. Also, we present similar result about the reversed hazardrate by the property of Fisher information in maxima.Some statistical data are most easily accessed in terms of record values. How much information

is contained in record observations? This question was addressed by Ahmadi and Arghami (2001,2003) and Hofmann (2004) by means of a comparison between the Fisher information in an iidsample and record data. They point out that for many distributions and parameters, the Fisherinformation in the first m record statistics is larger than the Fisher information in m iidobservations. In recent years, the problem of obtaining the Fisher information in record statisticswere studied by Hofmann and Nagaraja (2003), and Stepanov et al. (2003). In Section 3, weconjecture a characterization of the hazard function which is a product of a function of the

ARTICLE IN PRESS

G. Hofmann et al. / Statistics & Probability Letters 72 (2005) 51–57 53

variable and a function of the parameter using the property of Fisher information in the recordvalues.

2. Order statistics

Let X i:n be the ith order statistic from the random sample X 1; . . . ;X n from F ðx; yÞ: Denote thepdf, cdf and the hazard function of X i:n by f i:nðx; yÞ;Fi:nðx; yÞ and hi:nðx; yÞ; respectively. Then thehazard function of X 1:n is given by

h1:nðx; yÞ ¼f 1:nðx; yÞF1:nðx; yÞ

¼ nhðx; yÞ, (2)

thus the hazard function of X 1:n is n times that of X. By (1) and (2), we have the following result.

Lemma 2.1.

IðX 1:n; yÞ ¼Z þ1

�1

qqylog hðx; yÞ

� �2nðF ðx; yÞÞn�1f ðx; yÞdx. (3)

Because the hazard function of �X n:n is n times f ð�x; yÞ=F ð�x; yÞ; we have

IðX n:n; yÞ ¼Z þ1

�1

qqylog

f ðx; yÞF ðx; yÞ

� �2dFn:nðx; yÞ. (4)

In the following theorem, we weaken the condition provided by Zheng (2001) for the factorizationof the hazard function.

Theorem 2.2. The hazard function satisfies hðx; yÞ ¼ aðxÞaðyÞ for some functions aðxÞ and aðyÞ if andonly if, IðX 1:n; yÞ ¼ IðX ; yÞ for all nX1:

Proof. If hðx; yÞ ¼ aðxÞaðyÞ; it is easily shown that IðX 1:n; yÞ ¼ IðX ; yÞ for all nX1: Onthe other hand, if IðX 1:n; yÞ ¼ IðX ; yÞ for all nX1; by using (3), this condition can beexpressed as

IðX ; yÞZ þ1

�1

nðFðx; yÞÞn�1f ðx; yÞdx ¼ IðX ; yÞ by the assumption of Theorem

¼ IðX 1:n; yÞ

¼

Z þ1

�1

qqylog hðx; yÞ

� �2nðF ðx; yÞÞn�1f ðx; yÞdx.

Using the transformation u ¼ 1� F ðx; yÞ; it follows that for all nX1;Z 1

0

qqylog hðF�1ð1� uÞ; yÞ

� �2� IðX ; yÞ

" #un�1 du ¼ 0. (5)

ARTICLE IN PRESS

G. Hofmann et al. / Statistics & Probability Letters 72 (2005) 51–5754

It is well-known that fum;mX0g is a complete family of polynomials on ð0; 1Þ: Hence, (5) can onlyhold if

qqylog hðF�1ð1� uÞ; yÞ

� �2� IðX ; yÞ ¼ 0,

for all 0pup1: By transforming back to x, we have

qqylog hðx; yÞ

� �2� IðX ; yÞ ¼ 0,

for all x on the support of F. Since IðX ; yÞ does not depend on x, the result follows. &

It can easily be shown that hðx; yÞ ¼ aðyÞaðxÞ holds, if and only if

Fðx; yÞ ¼ 1� exp½�aðyÞAðxÞ�.

Thus, we have the following corollary.

Corollary 2.3. The family of distribution of the form F ðx; yÞ ¼ 1� exp½�aðyÞAðxÞ� can becharacterized by IðX 1:n; yÞ ¼ IðX ; yÞ for all nX1:

Remark. From a straightforward argument (Zheng, 2001), it can be seen that if y is a locationparameter or a scale parameter the representation of F implies that it is essentially of extremevalue type F ðx; yÞ ¼ 1� expð� expðsðx � yÞÞÞ; s40; or of Weibull type Fðx; yÞ ¼ 1�expð�ðx=yÞsÞ; s40; respectively. The equivalent statement holds for the representation F ðy; yÞ ¼e�bðyÞbðyÞ under the invariance of the FI for maxima. It should be mentioned that Gertsbakh andKagan (1999) characterized both the extreme value and the Weibull type using properties of theFisher information under Type I censoring.

If X is a random variable with an absolutely continuous distribution F, then the reversed hazardfunction of X at point t is defined as erðtÞ ¼ ðd=dtÞðlog F ðtÞÞ: One interpretation of the reversedhazard function at time t is the following. Suppose that X is nonnegative with distributionfunction F. Then X can be thought of as the lifetime of some device. Given that the device hasalready failed by time t, then the probability that it survived up to time t � � for a small �40 isapproximately �erðtÞ; see, Shaked and Shanthikumar (1994).The following result shows that the factorization of the reverse hazard function can be

characterized by the invariance of Fisher information in maxima.

Theorem 2.4. The reversed hazard function satisfies erðx; yÞ ¼ bðyÞbðxÞ for some functions bðxÞ andbðyÞ if and only if IðX n:n; yÞ ¼ IðX ; yÞ for all nX1:

Proof. The result follows from (4) by proceeding as in the proof of Theorem 2.2. &

Corollary 2.5. The family of distributions of the form F ðx; yÞ ¼ exp½�bðyÞBðxÞ� can becharacterized by IðX n:n; yÞ ¼ IðX ; yÞ for all nX1:

ARTICLE IN PRESS

G. Hofmann et al. / Statistics & Probability Letters 72 (2005) 51–57 55

3. Record values

Let fX i; iX1g be a sequence of iid continuous random variables with a cdf and pdf F ðx; yÞ andf ðx; yÞ; respectively. The random variable X n is called an upper (lower) record value of thissequence if X n4X i ðX noX iÞ for all i ¼ 1; 2; . . . ; n � 1: By convention X 1 is a record value. Formore details about record values we refer to Arnold et al. (1998). In recent years several authorshave studied the problem of Fisher information contained in record statistics, see, Ahmadi andArghami (2001, 2003), Hofmann and Nagaraja (2002) and Stepanov et al. (2003). We denote thenth upper (lower) record values by Un ðLnÞ: The marginal density of Un is given by

f Unðx; yÞ ¼

½� logð1� Fðx; yÞÞ�n�1

ðn � 1Þ!f ðx; yÞ, (6)

and the marginal density of Ln is given by

f Lnðx; yÞ ¼

½� log Fðx; yÞ�n�1

ðn � 1Þ!f ðx; yÞ.

In this section, we need the following lemma:

Lemma 3.1 (Zacks, 1971, p. 184). Let TðX 1; . . . ;X nÞ be a statistics, thenIðTðX 1; . . . ;X nÞ; yÞpIðX 1; . . . ;X n; yÞ; for all y 2 Y: Equality holds if and only if T is a sufficientstatistic.

It appears that the factorization of the hazard function (the reversed hazard function), into aproduct of a function of the variable and a function of the parameter, can be characterized by theequality of Fisher information contained in the nth upper (lower) record value and in a randomsample of size n from the original distribution.

Conjecture 3.2. The hazard function satisfies hðx; yÞ ¼ gðyÞcðxÞ for some functions cðxÞ and gðyÞ if

and only if IðUn; yÞ ¼ nIðX ; yÞ for all nX1:

Proof (Partial). If hðx; yÞ ¼ gðyÞcðxÞ; then from Corollary 3.1 of Ahmadi and Arghami (2001), wehave IðU1; . . . ;Un; yÞ ¼ nIðX ; yÞ for all nX1: Since by Theorem 3.2 of the same paper, among thefirst n upper record values, Un is a sufficient statistic, by Lemma 3.1 it follows that IðUn; yÞ ¼nIðX ; yÞ for all nX1: We currently cannot prove the reverse part. &

Corollary 3.3. The family of distributions of the form F ðx; yÞ ¼ 1� exp½�gðyÞCðxÞ� could be

characterized with the property of IðUn; yÞ ¼ nIðX ; yÞ for all nX1:

Given Conjecture 3.2 is true, similar results hold for the reverse hazard function when usinglower records.

4. Examples

The following examples give some applications of the results of this paper.(i) Systems of components: Let X 1;X 2; . . . be iid lifetimes of components of a system having a

common pdf f ðx; yÞ and cdf F ðx; yÞ; respectively. The lifetime of a series system with n

ARTICLE IN PRESS

G. Hofmann et al. / Statistics & Probability Letters 72 (2005) 51–5756

components is determined by X 1:n ¼ minfX 1; . . . ;X ng and the lifetime of a parallel system with n

components is determined by X n:n ¼ maxfX 1; . . . ;X ng:

Example 4.1. Consider a system with n components each having Burr type XIIðy; bÞ-distributionF ðx; yÞ ¼ 1� exp½�y logð1þ xbÞ�; where x; y40 and the shape parameter b is known. Then, wehave hðx; yÞ ¼ ybxb�1=ð1þ xbÞ: Without loss of generality we assume that b ¼ 1; which impliesIðX ; yÞ ¼ 1=y2: Thus by Theorem 2.2, IðX 1:n; yÞ ¼ 1=y

2 and by Conjecture 3.2, IðUn; yÞ ¼ n=y2 forall nX1: It can be shown that the Fisher information in X n:n is given by

IðX n:n; yÞ ¼

1

y2ð1þ 2nðn � 1Þðzð3Þ � 1ÞÞ for n ¼ 2;

1

y21þ 2nðn � 1Þ

Xn�3i¼0

n � 3

i

� �ð�1Þi

ð2þ iÞ3

!for nX3;

8>>>><>>>>:where z is the Riemann Zeta function. Therefore, IðX n:n; yÞXIðX ; yÞ; which means that in this casethe lifetime of a parallel system reveals more information about the parameter than the lifetime ofa serial system.(ii) Minimal repairs: Basically, it is said that a component with lifetime X and distribution

function Fðx; yÞ has been minimally repaired upon failure at time x0 if the distribution functionfor its next failure is given by ðF ðx; yÞ � F ðx0; yÞÞ=ð1� F ðx0; yÞÞ for xXx0: Let X ðnÞ denote thelifetime of the component if n minimal repairs are allowed, then

PðX ðnÞ4xÞ ¼ ð1� Fðx; yÞÞXn

i¼0

½� logð1� F ðx; yÞÞ�i

i!,

see, Shaked and Shanthikumar (1994). In view of (6), the right-hand side of above expression isthe survival function of the ðn þ 1Þth upper record value.

Example 4.2. Consider a component with lifetime X with Exponential distribution ½Fðx; yÞ ¼1� e�yx� and n minimal repairs are allowed. We have hðx; yÞ ¼ y and IðX ; yÞ ¼ 1=y2; then byConjecture 3.2, IðUn; yÞ ¼ n=y2; thus IðX ðnÞ; yÞ ¼ ðn þ 1Þ=y2 also, by Theorem 2.2, IðX 1:n; yÞ ¼1=y2:

Example 4.3. Let X be a random variable having the power distribution with cdf Fðx; yÞ ¼xy; 0pxp1; y40: We have erðx; yÞ ¼ y=x and IðX ; yÞ ¼ 1=y2; then by Theorem 2.4, IðX n:n; yÞ ¼1=y2; we also have IðLn; yÞ ¼ n=y2:

Acknowledgements

During the initial work, the first author was supported by Fondo Nacional de DesarrolloCientifico y Tecnologico (FONDECYT) Grant 1020479 of Chile. We are grateful to the refereefor various comments that let to improvements in the paper.

ARTICLE IN PRESS

G. Hofmann et al. / Statistics & Probability Letters 72 (2005) 51–57 57

References

Abo-Eleneen, Z.A., Nagaraja, H.N., 2002. Fisher information in an order statistic and its concomitant. Ann. Inst.

Statist. Math. 54, 667–680.

Ahmadi, J., Arghami, N.R., 2001. On the Fisher information in record values. Metrika 53, 195–206.

Ahmadi, J., Arghami, N.R., 2003. Comparing the Fisher information in record values and iid observations. Statistics

37, 435–441.

Arnold, B.C., Balakrishnan, N., Nagaraja, H.N., 1992. A First Course in Order Statistics. Wiley, New York.

Efron, B., Johnstone, I., 1990. Fisher information in terms of the hazard rate. Ann. Stat. 18, 38–62.

Gertsbakh, I., Kagan, A., 1999. Characterization of the Weibull distribution by properties of the Fisher information

under Type-I censoring. Statist. Probab. Lett. 42, 99–105.

Hofmann, G., 2004. Comparing Fisher information in record data and random observations. Statist. Papers 45,

517–528.

Hofmann, G., Nagaraja, H.N., 2003. Fisher information in record data. Metrika 57, 177–193.

Iyengar, S., Kvam, P., Singh, H., 1999. Fisher information in weighted distributions. Canad. J. Statist. 27, 833–841.

Lehmann, E.L., Casella, G., 1998. Theory of Point Estimation, second ed. Springer, New York.

Mehrotra, K.G., Johnson, R.A., Bhattacharyya, G.K., 1979. Exact Fisher information for censored samples and the

extended hazard rate functions. Comm. Statist.—Theory Methods 15, 1493–1510.

Park, S., 1996. Fisher information in order statistics. J. Amer. Statist. Assoc. 91, 385–390.

Rao, C.R., 1973. Linear Statistical Inference and its Applications, second ed. Wiley, New York.

Shaked, M., Shanthikumar, J.G., 1994. Stochastic Orders and their Applications. Academic Press, New York.

Stepanov, A.V., Balakrishnan, N., Hofmann, G., 2003. Exact distribution and Fisher information of weak record

values. Statist. Probab. Lett. 64, 69–81.

Zacks, S., 1971. The Theory of Statistical Inference. Wiley, New York.

Zheng, G., 2001. A characterization of the factorization of hazard function by the Fisher information under Type II

censoring with application to the Weibull family. Statist. Probab. Lett. 52, 249–253.

Zheng, G., Gastwirth, J.L., 2000. Where is the Fisher information in an ordered sample? Statist. Sinica 10, 1267–1280.

Zheng, G., Gastwirth, J.L., 2001. On the Fisher information in randomly censored data. Statist. Probab. Lett. 52,

421–426.