ARTICLE IN PRESS

0167-7152/$ - se

doi:10.1016/j.sp

�CorrespondE-mail addr

1Supported b

Statistics & Probability Letters 77 (2007) 1467–1472

www.elsevier.com/locate/stapro

On stochastic orderings between residual record values

Baha-Eldin Khaledia,b,�,1, Roohollah Shojaeia,b

aDepartment of Statistics, Shahid Beheshti University, Tehran, IranbRazi University, Kermanshah, Iran

Received 28 March 2006; received in revised form 17 December 2006; accepted 20 March 2007

Available online 7 April 2007

Abstract

In this paper we establish some stochastic ordering results among residual record values in two sample problems.

We also discuss some applications.

r 2007 Elsevier B.V. All rights reserved.

Keywords: k-records; Nonhomogeneous Poisson process; Minimal repair process; Gamma and beta distributions; Likelihood ratio;

Hazard rate; Usual stochastic and dispersive orderings

1. Introduction

Let fX i; iX1g be a sequence of i.i.d random variables from a continuous distribution F with hazard ratefunction rF , and let k be a positive integer. The random variables LðkÞðnÞ given by LðkÞð1Þ ¼ 1,

LðkÞðnþ 1Þ ¼ minfj 2 N;X j:jþk�14X LðkÞðnÞ:LðkÞððnÞþk�1Þg; nX1,

are called the nth k-record times and the quantities X LðkÞðnÞ:LðkÞððnÞþk�1Þ which we denote by RXn:k are termed the

nth k-records (cf. Kamps, 1995, p. 34).Let fZi; iX1g be a sequence of i.i.d. standard exponential random variables (with hazard rate equal to 1).

Then the joint density function of ðRZm:k;R

Zn:kÞ;mon, is

f ðRZm:k ;R

Zn:kÞðu; vÞ ¼

kn

GðmÞGðn�mÞum�1ðv� uÞn�m�1e�kv; upv, (1.1)

from which it follows that the density function of RZm:k is

f RZm:kðuÞ ¼

km

GðmÞum�1e�ku. (1.2)

e front matter r 2007 Elsevier B.V. All rights reserved.

l.2007.03.033

ing author.

ess: bkhaledi@hotmail.com (B.-E. Khaledi).

y Shahid Beheshti University, Tehran, Iran, Research Project 60-29-2005.

ARTICLE IN PRESSB.-E. Khaledi, R. Shojaei / Statistics & Probability Letters 77 (2007) 1467–14721468

Further, it is also known that

ðRXm:k;R

Xn:kÞ ¼

stðF�1ð1� e�RZ

m:k Þ;F�1ð1� e�RZn:k ÞÞ. (1.3)

For k ¼ 1 the k-records model reduces to the well-known ordinary (classic) record model. Nagaraja (1988)showed that

ðRXm:k;R

Xn:kÞ ¼

stðRVX

m:1;RVX

n:1 Þ, (1.4)

where the distribution function of V X is HðxÞ ¼ 1� FkðxÞ, from which it follows that the hazard rate of V X is

rH ðxÞ ¼ krF ðxÞ. (1.5)

For more details on distribution theory and various applications of records the reader is referred to Arnoldet al. (1998) and Kamps (1995).

In this paper, for any random variable X and an event A, ðX jAÞ denotes a random variable whosedistribution is the conditional distribution of X given A.

Let T be the lifetime to failure of a unit. Then given that the unit is of age t, the remaining life after time t,that is, ðT � tjT4tÞ, is random and called residual life random variable. The mean of this random variable iscalled mean residual life (MRL) at time t. For a comprehensive review on MRL function and its variousapplications in parametric as well as nonparametric contexts the reader is referred to Krishnan and Rao (1988)and references there in. In the following we consider new residual random variables for record values. Supposewe know that RX

m:k4t (RXm:k ¼ tÞ and we are interested in estimating or evaluating various characteristics of

the next records say RXn:k, n4m. Under this setting, it is more acceptable to consider either ðRX

n:k � tjRXm:k4tÞ or

ðRXn:k � tjRX

m:k ¼ tÞ as residual record value than ðRXn:k � tjRX

n:k4tÞ, since RXm:k4t as well as RX

m:k ¼ t impliesthat RX

n:k4t.There are several notions of stochastic orderings of varying degree of strength and they have been discussed

in detail in Shaked and Shanthikumar (1994). We first briefly review some of these here.Throughout this paper increasing means nondecreasing and decreasing means nonincreasing; and we shall be

assuming that all distributions under study are absolutely continuous. Let X and Y be univariate randomvariables with distribution functions F and G, survival functions F and G, density functions f and g, andhazard rates rF ð¼ f =F Þ and rG ð¼ g=GÞ, respectively. Let lX ðlY Þ and uX ðuY Þ be the left and the right endpointsof the support of X ðY Þ. X is said to be stochastically smaller than Y (denoted by XpstY ) if F ðxÞpGðxÞ for allx. This is equivalent to saying that EgðX ÞpEgðY Þ for any increasing function g for which expectations exist. X

is said to be smaller than Y in hazard rate ordering (denoted by XphrY ) if GðxÞ=F ðxÞ is increasing inx 2 ð�1;maxðuX ; uY ÞÞ. In case the hazard rates exist, it is easy to see that XphrY , if and only if, rGðxÞprF ðxÞ

for every x. X is said to be smaller than Y in the likelihood ratio order (and written as XplrY ) if gðxÞ=f ðxÞ isincreasing in x 2 ð�1;maxðuX ; uY ÞÞ. Note that likelihood ratio ordering implies hazard rate ordering which inturn implies stochastic ordering. Let F�1 and G�1 be the right continuous inverses (quantile functions) of F

and G, respectively. We say that X is less dispersed than Y (denoted by XpdispY ) if F�1ðbÞ � F�1ðaÞpG�1ðbÞ � G�1ðaÞ, for all 0papbp1. A consequence of XpdispY is that jX 1 � X 2jpstjY 1 � Y 2j and which inturn implies varðX ÞpvarðY Þ as well as E½jX 1 � X 2j�pE½jY 1 � Y 2j�, where X 1;X 2ðY 1;Y 2Þ are twoindependent copies of X ðY Þ.

In the next section we establish some stochastic ordering results among residual record values in two sampleproblems. We also discuss some applications.

2. Main results

Stochastic comparisons of record values have been considered in Kochar (1996), Raqab and Amin (1996),Ahmadi and Arghami (2001), Yue and Cao (2001), Belzunce and Shaked (2001), Belzunce et al. (2001, 2005),and Khaledi (2005) among others. Kochar (1990) proved that for random variable X with absolutelycontinuous distribution function F, RX

m:1plrRXn:1. Raqab and Amin (1996) extended the above result and

showed that for kX1, RXm:kplrR

Xn:k. For the case when X is DFR, Kochar (1996) showed that for mon, RX

m isless than RX

n for mon according to dispersive ordering. Ahmadi and Arghami (2001) and Belzunce et al.(2001) considered the stochastic comparisons among record values in two sample problems. They showed that

ARTICLE IN PRESSB.-E. Khaledi, R. Shojaei / Statistics & Probability Letters 77 (2007) 1467–1472 1469

XpstY ) RXn:1pstR

Yn:1 and XphrY ) RX

n:1phrRYn:1. With the help of a counterexample, they also show that

XplrY may not imply RXn:1plrR

Yn:1. Recently, Khaledi (2005) proved the following more general result.

Theorem 2.1. Let RXn:k,nX1, be a sequence of k-records corresponding to continuous distribution F and let RY

n0:k0 ,n0X1, be another sequence of k0-records corresponding to continuous distribution G, such that k0pk. Then, for

n0Xn,

(a)

XpstY ) RXn:kpstR

Yn0:k0 ,

(b)

XphrY ) RXn:kphrR

Yn0:k0 , and

(c)

if either X or Y is DFR, XpdispY ) RXn:kpdispRY

n0:k0 .

In the following we show that similar result holds when we compare two sequences of residual recordsðRX

n:k � tjRXm:k ¼ tÞ and ðRY

n:k � tjRYm:k ¼ tÞ. Let X t denote the residual random variable ðX � tjX4tÞ with

distribution function F t.

Theorem 2.2. Let RXn:k, nX1, be a sequence of k-records corresponding to continuous distribution F and let RY

n0:k0 ,n0X1, be another sequence of k0-records corresponding to continuous distribution G, such that kpk0. Then, for

n4m, n04m0, and n�mpn0 �m0,

(a)

XphrY ) ðRXn:k � tjRX

m:k ¼ tÞphrðRYn0:k0 � tjRY

m0:k0 ¼ tÞ and

(b)

if either X or Y is DFR, then

X tpdispY t ) ðRXn:k � tjRX

m:k ¼ tÞpdispðRYn0:k0 � tjRY

m0:k0 ¼ tÞ.

Proof. Using (1.1)–(1.3), it follows that

ðRXn:k � tjRX

m:k ¼ tÞ¼stRX t

n�m:k,

where RX t

n�m:k is the ðn�mÞth k-record corresponding to random variable X t. On the other hand,X tphrY t3XphrY and, X is DFR if and only if X t is DFR. Combining these observations, the requiredresults follow from Theorem 2.1. &

To prove the next result in this section we use the following lemma whose proof is elementary and henceomitted.

Lemma 2.1. Let V1 and V2 be two Poisson random variables with parameters l1 and l2, respectively. If l1Xl2then ðV 1jV1pmÞXlrðV 2jV2pmÞ, m ¼ 0; 1; . . . :

Theorem 2.3. Let RXn:1, nX1, be a sequence of 1-records corresponding to continuous distribution F and let RY

n:1,nX1, be another sequence of 1-records corresponding to continuous distribution G. Then, for mon,

XphrY ) ðRXn:1 � tjRX

m:14tÞpstðRYn:1 � tjRY

m:14tÞ. (2.1)

Proof. Using (1.1)–(1.3), for fixed mpn and t, the distribution function of ðRXn:1 � tjRX

m:14tÞ for xX0 can bewritten as

pðRXn:1pxþ tjRX

m:14tÞ ¼pðtoRX

m:1pRXn:1pxþ tÞ

pðRXm:14tÞ

¼pð� lnF ðtÞoRZ

m:1pRZn:1p� lnF ðxþ tÞÞ

pðRZm:14� lnF ðtÞÞ

¼

R� lnF ðxþtÞ

� lnF ðtÞ

R v

� lnF ðtÞðe�vðv� uÞn�m�1um�1=GðmÞGðn�mÞÞdudvR1� lnF ðtÞ

ðum�1e�u=GðmÞÞdu

ARTICLE IN PRESSB.-E. Khaledi, R. Shojaei / Statistics & Probability Letters 77 (2007) 1467–14721470

¼

R� lnF ðxþtÞ

� lnF ðtÞðe�v=GðnÞÞ

R 1� lnF ðtÞ=v

ððv� svÞn�m�1ðsvÞm�1=bðm; n�mÞÞvdsdvR1

� lnF ðtÞðum�1e�u=GðmÞÞdu

¼

R� lnF ðxþtÞ

� lnF ðtÞðvn�1e�v=GðnÞÞ

R 1� lnF ðtÞ=v

ðð1� sÞn�m�1sm�1=bðm; n�mÞÞdsdvR1� lnF ðtÞ

ðum�1e�u=GðmÞÞdu. ð2:2Þ

Binomial expansion for beta distribution function simplifies (2.2) as

R� lnF ðxþtÞ

� lnF ðtÞðvn�1e�v=GðnÞÞ

Pm�1i¼0

n� 1

i

� �ð� lnF ðtÞ=vÞið1� ð� lnF ðtÞ=vÞÞn�i�1 dv

R1� lnF ðtÞ

ðum�1e�u=GðmÞÞdu. (2.3)

Now, using Poisson expansion for gamma distribution function, (2.3) would be

Pm�1i¼0

n� 1

i

� �ðð� lnF ðtÞÞi=GðnÞÞ

R� lnF ðxþtÞ

� lnF ðtÞe�vðvþ lnF ðtÞÞn�i�1 dv

Pm�1j¼0 ð� lnF ðtÞÞje�ð� lnF ðtÞÞ=j!

. (2.4)

After some simplifications, (2.4) can be written as

Pm�1i¼0 ðð� lnF ðtÞÞi=i!Þe�ð� lnF ðtÞÞPðGn�ioð� lnF ðxþ tÞ=F ðtÞÞÞPm�1

j¼0 ð� lnF ðtÞÞje�ð� lnF ðtÞÞ=j!, (2.5)

where Gn�i is gamma random variable with shape parameter n� i and scale parameter 1.Let V 1 be a Poisson random variable with parameter ð� lnF ðtÞÞ and let pF ðiÞ ¼ pðV1 ¼ ijV1pm� 1Þ,

m ¼ 1; 2 . . . : Then

pðRXn:1pxþ tjRX

m:14tÞ ¼Xm�1i¼0

pF ðiÞP Gn�ip � lnF ðxþ tÞ

F ðtÞ

� �� �. (2.6)

Similarly, the distribution function of ðRYn:1 � tjRY

m:14tÞ can be written as

pðRYn:1pxþ tjRY

m:14tÞ ¼Xm�1i¼0

pGðiÞP Gn�ip � lnGðxþ tÞ

GðtÞ

� �� �, (2.7)

where pGðiÞ ¼ pðV2 ¼ ijV2pm� 1Þ in which V2 is Poisson random variable with parameter ð� lnGðtÞÞ. Theassumption X 1phrY 1 implies that X 1pstY 1. That is ð� lnF ðtÞÞXð� lnGðtÞÞ. Thus, by Lemma 2.1 and the factthat likelihood ratio order implies usual stochastic order we get

ðV 1jV1pm� 1ÞXstðV2jV 2pm� 1Þ. (2.8)

The assumption X 1phrY 1 is equivalent to

Gðxþ tÞ

GðtÞX

F ðxþ tÞ

F ðtÞ; x40.

Therefore,

P Gn�ip � lnF ðxþ tÞ

F ðtÞ

� �� �XP Gn�ip � ln

Gðxþ tÞ

GðtÞ

� �� �. (2.9)

As a final observation the random variable Gn�i is decreasing in i with respect to usual stochastic order.That is,

PðGn�ipxÞ is increasing function of i. (2.10)

ARTICLE IN PRESSB.-E. Khaledi, R. Shojaei / Statistics & Probability Letters 77 (2007) 1467–1472 1471

Combining these observations we have for x40 that

pðRXn:1pxþ tjRX

m:14tÞ ¼Xm�1i¼0

pF ðiÞP Gn�ip � lnF ðxþ tÞ

F ðtÞ

� �� �

X

Xm�1i¼0

pF ðiÞP Gn�ip � lnGðxþ tÞ

GðtÞ

� �� �by ð2:9Þ

X

Xm�1i¼0

pGðiÞP Gn�ip � lnGðxþ tÞ

GðtÞ

� �� �; by ð2:8Þ and ð2:10Þ.

This proves the required result. &

In the next theorem we extend the result of Theorem 2.3 from 1-records to k-records.

Theorem 2.4. Let RXn:k, nX1, be a sequence of k-records corresponding to continuous distribution F and let RY

n:k0 ,nX1, be another sequence of k0-records corresponding to continuous distribution G. Then, for mpn and kXk0,

XphrY ) ðRXn:k � tjRX

m:k4tÞpstðRYn:k0 � tjRY

m:k04tÞ. (2.11)

Proof. Using assumptions kXk0, rF ðxÞXrGðxÞ, and relations (1.4) and (1.5), the required result follows fromTheorem 2.3. &

3. Applications

There is a close connection between record values and the epoch times of nonhomogeneous Poisson process(NHPP). A counting process fNðtÞ; tX0g is NHPP with rate function rðtÞX0 if it satisfies the followingassumptions:

(a)

Nð0Þ ¼ 0, (b) fNðtÞ; tX0g has independent increments, (c) pfNðtþ hÞ �NðtÞX2g ¼ oðhÞ, (d) pfNðtþ hÞ �NðtÞ ¼ 1g ¼ rðtÞhþ oðhÞ.

Let Si, iX1, be the time of the ith event of the NHPP, then, for mon,

ðSm;SnÞ ¼stðF�1ð1� e�S�m Þ;F�1ð1� e�S�n ÞÞ, (3.1)

where S�i , iX1, is the time of the ith event of Poisson process with constant rate function equal to 1 and F is anabsolutely continuous distribution function with hazard rate rðtÞ.

Now let NðtÞ denote the number of ordinary record values less than or equal to t. Comparing (1.3) and (3.1),it follows that NðtÞ is an counting process of events where an event is said to occur at time x if x is an ordinaryrecord value. In this case NðtÞ is an NHPP with rate function rF and ordinary record values are epoch times ofthis NHPP. Further, if we define NðtÞ to be the number of k-records less than or equal to t, it follows from(1.5) that NðtÞ, in this case, is also an NHPP with rate k rF . For more details on the NHPP see Ross (1997,p. 277). Suppose we know that either Sm4t or Sm ¼ t and we are interested in estimating or evaluating variouscharacteristics of the next epoch time Sn, n4m. Since the sequence fSi; iX1g is stochastically the same as thesequence fRX

i:k; iX1g, then results obtained in this section could be applied to any field of probability andstatistics where NHPP is used. For example, in reliability theory it appears in the context of minimal repair ofan item. The times that minimal repairs are made are distributed according to the epoch times of an NHPP.

ARTICLE IN PRESSB.-E. Khaledi, R. Shojaei / Statistics & Probability Letters 77 (2007) 1467–14721472

Let fZli ; iX1g be a sequence of i.i.d exponential random variables with common hazard rate l. Then,

EðRZl

n:k � tjRZl

m:k4tÞ ¼

Z 10

Xm�1i¼0

ðtklÞi=i!Pm�1j¼0 ðtklÞ

j=j!PðGn�i4kxlÞdx

¼Xm�1i¼0

ðtklÞi=i!Pm�1j¼0 ðtklÞ

j=j!

Z 10

PðGn�i4kxlÞdx

¼Xm�1i¼0

ðtklÞi=i!Pm�1j¼0 ðtklÞ

j=j!E

Gn�i

kl

� �

¼Xm�1i¼0

ðtklÞi=i!Pm�1j¼0 ðtklÞ

j=j!

n� i

kl. ð3:2Þ

For arbitrary F (known or unknown), in many cases, various characteristics of ðRXn:k � tjRX

m:k4tÞ cannot beeasily computed. In the following we will obtain a convenient lower and upper bounds on the mean of the newresidual records. Let fX i; iX1g be a sequence of i.i.d. random variables such that the common hazard rate ofX i’s, denoted by rðtÞ, is not completely known, but it is known that for all t, lLprðtÞplU. The fact that hazardrate of X i’s is bounded from above by lU means that ZlU

1 phrX 1 and is bounded from below by lL means thatX 1phrZ

lL1 . Now using (3.2) it follows from Theorem 2.4 that

Xm�1i¼0

ðtklUÞi=i!Pm�1

j¼0 ðtklUÞj=j!

n� i

klUpEðRX

n:k � tjRXm:k4tÞp

Xm�1i¼0

ðtklLÞi=i!Pm�1

j¼0 ðtklLÞj=j!

n� i

klL.

In particular, let fX i; iX1g be a sequence of i.i.d. gamma random variables with common shape parametera41 and mean a=l, l40. Then it is known that Zl

1phrX 1, therefore using the above result we obtain a lowercomputable bound for the mean residual record EðRX

n:k � tjRXm:k4tÞ.

Acknowledgement

The authors are grateful to the referee for his constructive comments and suggestions.

References

Ahmadi, J., Arghami, N.R., 2001. Some univariate stochastic orders on record values. Comm. Statist. Theory Methods 30, 69–78.

Arnold, B.C., Balakrishnan, N., Nagaraja, H.N., 1998. Records. Wiley, New York.

Belzunce, F., Shaked, M., 2001. Stochastic comparisons of mixtures of convexly ordered distributions with applications in reliability

theory. Statist. Probab. Lett. 53, 363–372.

Belzunce, F., Lillo, R.E., Ruiz, J.M., Shaked, M., 2001. Stochastic comparisons of nonhomogeneous processes. Probab. Eng. Statist. Sci.

15, 199–224.

Belzunce, F., Mercader, J.-A., Ruiz, J.-M., 2005. Stochastic comparisons of generalized order statistics. Probab. Eng. Statist. Sci. 19,

99–120.

Kamps, U., 1995. A Concept of Generalized Order Statistics. Teubner, Stuttgart.

Khaledi, B.E., 2005. Some new results on stochastic orderings between generalized order statistics. J. Iranian Statist. Soc. 4, 35–49.

Kochar, S.C., 1990. Some partial ordering results on record values. Comm. Statist. Theory Methods 19, 299–306.

Kochar, S.C., 1996. A note on dispersive ordering of record values. Calcutta Statist. Assoc. Bull. 46, 63–67.

Krishnan, P.R., Rao, C.R., 1988. Handbook of Statistics, vol. 7, Elsevier, Amsterdam, pp. 215–224.

Nagaraja, H.N., 1988. Record values and related statistics—a review. Comm. Statist. Theory Methods 17, 2223–2238.

Raqab, M.Z., Amin, W.A., 1996. Some ordering results on order statistics and record values. IAPQR Trans. 21 (1), 1–8.

Ross, S.M., 1997. Introduction to Probability Models. Academic Press, Harcourt Asia Pte Ltd., New York.

Shaked, M., Shanthikumar, J.G., 1994. Stochastic Orders and Their Applications. Academic Press, San Diego, CA.

Yue, D., Cao, J., 2001. Some results on successive failure times of a system with minimal instantaneous repairs. Oper. Res. Lett. 29,

193–197.