MetrikaDOI 10.1007/s00184-014-0485-9

On extremes of bivariate residual lifetimes fromgeneralized Marshall–Olkin and time transformedexponential models

Yinping You · Xiaohu Li ·Narayanaswamy Balakrishnan

Received: 21 June 2013© Springer-Verlag Berlin Heidelberg 2014

Abstract We study here extremes of residuals of the bivariate lifetime and the residualof extremes of the two lifetimes. In the case of generalized Marshall–Olkin modeland the total time transformed exponential model, we first present some sufficientconditions for the extremes of residuals to be stochastically larger than the residual ofthe corresponding extremes, and then investigate the stochastic order of the residualof extremes of the two lifetimes based on the majorization of the age vector of theresiduals.

Keywords GMO model · Marjorization · RTI · Stochastic order · TTE model

1 Introduction

For a bivariate nonnegative random vector X = (X1, X2) representing the two life-times or risks and the real vector t = (t1, t2) with t1, t2 ≥ 0, it is often of interest tostudy the probability behavior of X t = [X1 − t1, X2 − t2 | X1 > t1, X2 > t2], theresidual life vector of X at age t , for t1, t2 ≥ 0. In practice, the evolution of the prob-ability of the residual lifetime vector usually plays an important role in evaluating the

Supported by National Natural Science Foundation of China (11171278).

Y. You · X. Li (B)School of Mathematical Sciences, Xiamen University, Xiamen 361005, Chinae-mail: xhli@lzu.edu.cn; mathxhli@hotmail.com

N. BalakrishnanDepartment of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada

N. BalakrishnanDepartment of Statistics, King Abdulaziz University, Jidda, Saudi Arabia

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performance of the extreme of the two random variables. There are many applicationswhere this is naturally of interest including the following areas:

• In reliability engineering, suppose there are two lifetimes X1 and X2 for one com-plex system; for example, the natural lifetime and the flying lifetime of an airplane.Then, for t1, t2 ≥ 0, X t gives the residual lifetimes of the system given that theyare good at time t . In practice, such a system usually fails if one of the two lifetimesends, that is, the total lifetime is T = min{X1, X2}. If the system is found to begood after a regular preventive maintenance at time t, (Xi )ti = [Xi − ti | Xi > ti ](i = 1, 2) gives the two residual lifetimes, and the residual lifetime of the systemat age t is then min{X1 − t1, X2 − t2 | X1 > t1, X2 > t2}; here, t1 need not beequal to t2. In this situation, X1 and X2 are generally assumed to be dependent.Most studies in this direction focus on coherent systems with mutually independentcomponents at the one dimensional age time t . We refer the readers to Li and Lu(2003), Pellerey and Petakos (2002), Samaniego et al. (2009) and Li et al. (2013) forsome recent advances in this line of research. Moreover, for an engineer evaluatinga series system of two used components with an age vector, it will be of interest tostudy the system’s lifetime with respect to the majorization of the age vector.

• In insurance and actuarial science, suppose X1 and X2 are the potential total claimamounts of two dependent risks covered by an insurance policy. For example, in avi-ation insurance, a survived passenger may suffer from two different damages, whichresult in corresponding claims X1 and X2 with deductibles t1 and t2, respectively.Sometimes, the airline’s insurer is liable only for the larger damage to the insured,i.e., the potential loss is max{X1 − t1, X2 − t2 | X1 > t1, X2 > t2}. In order toarrange the budget or to allocate the risk capital, the insurer has to predict the poten-tial claim amounts based on the deductible vector (t1, t2). On the other hand, insurermay let the insured allocate a given amount of deductible t̄ = t1 + t2 to the two risksaccording to his/her own will. In such a situation, majorization of allocation policiess = (s1, s2) and t = (t1, t2) always makes sense, and thus a better understandingon the probability behavior of the potential loss with respect to the majorizationof the allocation policy t will help in coming up with an economic budget or areasonable allocation of risk capital or deductible. For some recent advances inrisk capital allocation, we refer the readers to Laeven and Goovaerts (2004), Buchand Dorfleitner (2008), Dhaene et al. (2012) and the references therein, and fordeductible allocation, one may refer to Li and You (2012) and Cheung (2007).

• In network security, the traffic monitoring and estimation of flow parameters inhigh-speed routers have become increasingly challenging as the Internet grows inboth scale and complexity. In order to scale measurement algorithms to achieveestimation while satisfying real-time resource constraints of high-speed Internetrouters (such as fixed memory consumption and per-packet processing delay), thesecurity analyst usually reduces the amount of information to be stored by a routerin its internal tables at the expense of deploying special estimation techniques thatcan recover metrics of interest from the sample collected on line; see, for example,Wang et al. (2013) and the references therein. In the residual sampling, each flowis truncated by a predetermined number and only the residual flow size is observedand an unbiased estimation of the flow size may be obtained based on the residuals.

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For example, the maximum likelihood estimator max{X1 − t1, . . . , Xn − tn | X1 >

t1, . . . , Xn > tn} is just the maximum of the observed residual vector. In such acontext, one can gain more accuracy in statistical estimation, while keeping memoryconsumption as small as possible, by studying the distribution of the maximum ofresiduals with truncation times t1, . . . , tn . Moreover, for a given total amount oftruncated information t̄ = t1 +· · ·+ tn , it will also be of interest to discuss how thetruncation plan vector (t1, . . . , tn) impacts the total amount of observed informationthrough the majorization order.

In the present work, we focus on the extremes of residual lifetimes of two componentshaving the generalized Marshall–Olkin model and the time transformed exponentialmodel, and obtain some sufficient conditions for the majorization order of the agevector to imply a stochastically larger extreme of the residuals. Section 2 recallssome pertinent concepts and notions as preliminaries. Section 3 conducts comparisonsbetween the residual of extremes and the extreme of residuals, and stochastic orderson the extremes of residuals from components following the generalized Marshall–Olkin model are studied in Sect. 4. In Sect. 5, we give some sufficient conditions forthe stochastic order on the extremes of residuals from components following the timetransformed exponential model, and Sect. 6 presents stochastic orders on the extremesof residuals from independent and identically distributed components. Finally, someconcluding remarks are made in Sect. 7.

2 Preliminaries

For ease of reference, we recall here some notions that are used in the sequel. Onemay refer to Shaked and Shanthikumar (2007) for more details.

A real vector s = (s1, s2) is said to be majorized by another vector t = (t1, t2)(denoted by s �m t) if s1 + s2 = t1 + t2 and max{s1, s2} ≤ max{t1, t2}. A randomvariable X is said to be stochastically smaller than another variable Y (denoted byX ≤st Y ) if P(X > x) ≤ P(Y > x) for all x . By the use of stochastic order, one maydefine two aging properties as follows: a nonnegative random variable X is said to beof increasing failure rate (IFR) if Xs ≥st Xt for all t ≥ s ≥ 0. As a natural dual notion,the decreasing failure rate (DFR) may be defined by reversing the above inequality.For more recent advances on stochastic orders with applications in reliability and risk,we refer the readers to Li and Li (2013).

A random vector X = (X1, X2) = (min{S1, S3}, min{S2, S3}

)is said to have the

generalized Marshall–Olkin (GMO) distribution if the generators S1, S2 and S3 aremutually independent lifetimes. Let Ri denote the cumulative hazard of Si , i = 1, 2, 3.Then, X has the survival function

F̄(x, y) = exp{ − R1(x) − R2(y) − R3(max{x, y})}.

The GMO model includes the bivariate exponential distribution (Marshall and Olkin1967), and Marshall–Olkin type distribution (Muliere and Scarsini 1987) as specialcases. For more details on GMO distributions, one may refer to Li and Pellerey (2011)and Lin and Li (2012).

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Barlow and Mendel (1992) and Barlow and Spizzichino (1993) were among the firstto study the following semi-parametric model: Two exchangeable random lifetimes(X1, X2) is said to follow the time transformed exponential (TTE) model if it hasthe joint survival function F̄(x, y) = W (R(x) + R(y)) (x ≥ 0, y ≥ 0) for somecontinuous, convex and strictly decreasing survival function W and a continuous,strictly increasing function R with R(0) = 0 and R(∞) = ∞. Obviously, W servesas the time transform and R just as the accumulated hazard. As a semi-parametricmodel, the TTE distribution provides a reasonable way to describe bivariate randomlifetimes with their dependence and aging property being separable. Incidentally, theTTE model corresponds to the bivariate distribution with an Archimedean copula,and is quite flexible in modeling bivariate dependent random risks in actuarial scienceand some other applied areas. In fact, it can be easily verified that the independentlaw corresponding to W (t) = e−λt for λ > 0, Schur-constant law correspondingto R(t) = t , and the frailty model corresponding to W being the Laplace transformof the frailty factor, are all special cases of TTE models. For some recent work inthis direction, we refer the readers to Bassan and Spizzichino (2001), Bassan andSpizzichino (2005) and Li and Lin (2011).

From now on, for the sake of convenience and without loss of generality, we alwaysassume t1 ≤ s1 ≤ s2 ≤ t2 whenever s = (s1, s2) �m (t1, t2) = t , and we shall denote1 = (1, 1) and

min{X − t} = min{X1 − t1, X2 − t2}, max{X − t} = max{X1 − t1, X2 − t2}.

In addition, we also adopt the notation t1 = (t, t), t − s = (t1 − s1, t2 − s2) andX > t = (X1 > t1, X2 > t2). Furthermore, throughout this article, we shall use theterms increasing and decreasing for non-decreasing and non-increasing, respectively.

3 Residual of extremes

When X1 and X2 are independent, Li and Lu (2003) carried out a stochastic com-parison between the extremes of [X − t1 | X > t1] and the residual life of thecorresponding extremes of (X1, X2). For example, max{(X1)t , (X2)t } was shown tobe stochastically larger than [max{X1, X2} − t | max{X1, X2} > t] for any t ≥ 0.In an independent investigation, Pellerey and Petakos (2002) considered this issuefor coherent systems with more than two components and proved that a coherentsystem with mutually independent used components having a common age is sto-chastically larger than the corresponding used coherent system with the same age.For a brief review of results in this direction, one may refer to Li and Ding (2013).Recently, Li et al. (2013) established some sufficient conditions on the dependence ofthe components for this stochastic ordering to hold. Here, we deal with the comparisonbetween the residuals of extremes and the extremes of residuals in the case of dependentlifetimes.

First of all, since for all t ≥ 0 and x ≥ 0,

P(

min{X − t1 | X > t1} > x) = P

(min{X} − t > x | min{X} > t

),

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we have

[min{X − t1 | X > t1}] st= [

min{X} − t | min{X} > t].

In reliability terms, the above stochastic equivalence asserts that the lifetime of a usedseries system is stochastically equal to that of the series system of used componentswith the same age whatever the dependence between the two components is.

For a bivariate random vector X = (X1, X2), X2 is said to be right tail increasing(RTI) in X1 if [X2 | X1 > x1] is stochastically increasing in x1. We can then establishthe following result for the maximum of two lifetimes having the TTE model.

Theorem 1 Suppose (X1, X2) follows the TTE model. If W is log-convex, then, fort ≥ 0,

max{X − t1 | X > t1} ≥st[

max{X} − t | max{X} > t]. (1)

Proof It is easy to verify that

P(X1 > x | X2 > y) = W (R1(x) + R2(y))/W (R2(y))

for x ≥ 0 and y ≥ 0. Since W is log-convex, we have W (x+a)W (x)

to be increasingin x for a ≥ 0. On the other hand, the increasing property of R2 guarantees thatR2(y2) ≥ R2(y1) for y2 ≥ y1 ≥ 0. So, for x ≥ 0, we have

W (R1(x) + R2(y2))/W (R2(y2)) ≥ W (R1(x) + R2(y1))/W (R2(y1)).

Consequently, we obtain, for x ≥ 0 and y2 ≥ y1 ≥ 0,

P(X1 > x | X2 > y2) ≥ P(X1 > x | X2 > y1),

meaning that X1 is RTI in X2.Likewise, X2 may also be proved to be RTI in X1. By Theorem 11.2.4 of Li et al.

(2013), we then get the desired stochastic order in (1). ��In reliability, (1) tells that the lifetime of a used parallel system with two components’

lifetimes having TTE model with a log-convex W is stochastically larger than that ofa parallel system of used components with the same age. Example 1 below asserts thatviolation of the assumption of log-convex property may undermine Theorem 1, and soassertion in (1) is not true for any dependence structure between the two components.

Example 1 (Frank copula) Assume (X1, X2) has a TTE model with, for θ �= 0,

W (x) = −1

θln

(1 + (e−θ − 1)e−x) and R1(x) = R2(x) = − ln

e−θe−x − 1

e−θ − 1.

Then, the joint survival function F̄ of (X1, X2) is given by

F̄(x, y) = −1

θln

(1 + (e−θ − 1)−1(e−θe−x − 1)(e−θe−y − 1)

), θ �= 0.

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0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.4

0.6

0.8

1.0

Fig. 1 Survival curves of Ut (solid) and Vt (dashed) for t = 1 and x ∈ (0, 1.5)

Denote Ut = max{X − t1 | X > t1}] and Vt = [max{X} − t | max{X} > t

]for

t ≥ 0. It is then easy to verify that their survival functions are given by

F̄Ut (x) = [2W

(R(x + t) + R(t)

) − W(2R(x + t)

)]/W

(2R(t)

),

F̄Vt (x) = [2W

(R(x + t)

) − W(2R(x + t)

)]/[2W

(R(t)

) − W(2R(t)

)],

respectively. Upon setting θ = −5 in the above expressions of W and R, it is easy tosee that

h(x) = W (x + 1)

W (x)= ln

(1 + (e5 − 1)e−x−1

)

ln(1 + (e5 − 1)e−x

) .

Since h(2) = 0.697197 < 0.758355 = h(1), h(x) is not increasing in x ; that is, Wdoes not possess the log-convex property. As seen in Fig. 1, F̄Ut (x) and F̄Vt (x) crosseach other for 0 ≤ x ≤ 1.5 and t = 1, which clearly invalidates the stochastic orderresult in Theorem 1. ��

4 Extremes of residuals of lifetimes following GMO models

In this section, we establish some sufficient conditions for the majorization of the agevector to imply the stochastic order of the minimum and maximum of the residuallifetimes.

Theorem 2 For a bivariate GMO vector X with generators S1, S2, S3, if S1 and S2have a common concave (convex) hazard rate and S3 is DFR (IFR), then s �m timplies

min{X − t | X > t} ≥st (≤st) min{X − s | X > s}.

Proof Note that s �m t implies t2 − s2 = s1 − t1 and t2 > s2. Let R1 = R2 = R.Since S1 and S2 have a common concave hazard rate, R(t + x) − R(t) is concave int for any x ≥ 0. Thus, we have, for any x ≥ 0,

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[R(t2 + x) − R(t2)] − [R(s2 + x) − R(s2)]≤ [R(s1 + x) − R(s1)] − [R(t1 + x) − R(t1)],

or equivalently, for x ≥ 0,

exp{

R(t2) + R(t1) − R(t2 + x) − R(t1 + x)}≥ exp

{R(s2) + R(s1) − R(s2 + x) − R(s1 + x)}. (2)

Since S3 is DFR, R3 is concave, and so R3(t2 + x) − R3(t2) ≤ R3(s2 + x) − R3(s2)

for s2 ≤ t2 and any x ≥ 0. Consequently, we have

exp{

R3(t2 + x) − R3(t2)} ≤ exp{

R3(s2 + x) − R3(s2)}. (3)

Upon combining (2) and (3), we get

exp{

R(t1) + R(t2) − R(t1 + x) − R(t2 + x)}/exp

{R3(t2 + x) − R3(t2)

}

≥ exp{

R(s1) + R(s2) − R(s1 + x) − R(s2 + x)}/exp

{R3(s2 + x) − R3(s2)

}.

Now, for x, t1 and t2 ≥ 0, since

P(min{X − t | X > t} > x)

= P(X1 > t1 + x, X2 > t2 + x)/P(X1 > t1, X2 > t2)

= exp{−R(t1 + x) − R(t2 + x) − R3(t2 + x)}/exp{−R(t1) − R(t2) − R3(t2)}= exp

{R(t1) + R(t2) − R(t1 + x) − R(t2 + x)

}/exp

{R3(t2 + x) − R3(t2)

},

we readily have P(min{X − t | X > t} > x) ≥ P(min{X − s | X > s} > x) forx ≥ 0, which is the desired result.

The other part of the result may be completed in an analogous manner. ��As seen in Example 2 below, the concavity of the hazard rate in Theorem 2 can not

be dropped when S3 is DFR.

Example 2 Let S1 and S2 have a common cumulative hazard rate R(x) = 16 x3+ 64

3 x3/2

and S3 have the standard exponential distribution, i.e., R3(x) = x . Then, for s =(6, 8) �m (2, 12) = t , we have

P(min{X − t | X > t} > x) = exp{

R(12) + R(2) − R(12 + x) − R(2 + x) − x},P(min{X − s | X > s} > x) = exp

{R(8) + R(6) − R(8 + x) − R(6 + x) − x}.

As can be seen in Fig. 2a, the survival curve corresponding to t lies below thatcorresponding to s, which invalidates the result min{X − t | X > t} ≥st min{X − s |X > s}. ��Theorem 3 For a bivariate GMO vector X with generators S1, S2, S3, if S1 andS2 have a common exponential distribution and S3 is DFR, then s �m t impliesmax{X − t | X > t} ≥st max{X − s | X > s}.

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1.0

0.9

0.8

0.7

0.6

0.0005 0.0010 0.0015 0.0020 0.0025 0.0030

0.6

0.8

1.0

0.4

0.2

0.5 1.0 1.5 2.0 2.5 3.0

(a) (b)

Fig. 2 Survival curves of the extreme of residuals at t (dashed) and s (solid) for GMO models. a Theminimum with s = (6, 8) and t = (2, 12). b The maximum with s = (3, 5) and t = (1, 7)

Proof Let λ > 0 be the common hazard rate of S1 and S2. Since S3 is DFR, itscumulative hazard R3 is concave. Then, R3(t)−R3(t+x) and so P(S3 > t+x |S3 > t)= exp{R3(t) − R3(t + x)} is increasing in t for x ≥ 0. Thus, for t2 ≥ s2, we have

exp{R3(t2) − R3(t2 + x)} ≥ exp{R3(s2) − R3(s2 + x)} (4)

and

P(S3 > x + t1 | S3 > t2) (5)

= P(S3 > x + t1 − t2 + t2 | S3 > t2)

≥ P(S3 > x + t1 − t2 + s2 | S3 > s2)

≥ P(S3 > x + s1 | S3 > s2) (t1 − t2 + s2 ≤ s1). (6)

Upon combining (4) and (6), we get, for x ≥ 0,

e−λx P(S3 > t1 + x | S3 > t2) + (e−λx − e−2λx) exp{R3(t2) − R3(t2 + x)}

≥ e−λx P(S3 > s1 + x | S3 > s2) + (e−λx − e−2λx) exp{R3(s2) − R3(s2 + x)}.

(7)

Due to the mutual independence among S1, S2 and S3, we have, for x ≥ 0,

P(X1 > t1 + x, X2 > t2 + x)

P(X1 > t1, X2 > t2)

= P(S1 > t1 + x, S2 > t2 + x, S3 > t2 + x)

P(S1 > t1, S2 > t2, S3 > t2)

= e−2λx exp{R3(t2) − R3(t2 + x)}

and P(X1 > t1 + x | X1 > t1, X2 > t2)

= P(S1 > t1 + x, S3 > t1 + x | Si > ti , i = 1, 2, 3)

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= P(S1 > t1 + x | S1 > t1)P(S3 > t1 + x | S3 > t2)

= e−λx P(S3 > t1 + x | S3 > t2).

Then, for x ≥ 0, we have

P(max{X − t | X > t} > x)

= P(X1 > t1 + x | X1 > t1, X2 > t2) + P(X2 > t2 + x | X1 > t1, X2 > t2)

−P(X1 > t1 + x, X2 > t2 + x | X1 > t1, X2 > t2)

= e−λx P(S3 > t1 + x | S3 > t2) + (e−λx − e−2λx ) exp{R3(t2) − R3(t2 + x)}.

As a consequence, (7) reduces to P(max{X − t | X > t} > x) ≥ P(max{X − s |X > s} > x) for all x ≥ 0, which completes the proof. ��

The following example shows that the DFR property of S3 in Theorem 3 is necessary.

Example 3 Let S1 and S2 have the standard exponential distribution and S3 have itscumulative hazard as R3(x) = 1

2 x2 + 4x1/2. Set s = (3, 5) �m (1, 7) = t . Then,

P(max{X1 − t1, X2 − t2 | X1 > t1, X2 > t2} > x)

={

e−x + [e−x − e−2x

]exp

{R3(7) − R3(7 + x)

}, if 0 < x ≤ 6,

exp{

R3(7) − R3(1 + x) − x} + (

e−x − e−2x)

exp{

R3(7) − R3(7 + x)}, if 6 < x,

and

P(max{X1 − s1, X2 − s2 | X1 > s1, X2 > s2} > x)

={

e−x + [e−x − e−2x

]exp

{R3(5) − R3(5 + x)

}, if 0 < x ≤ 2,

exp{

R3(5) − R3(3 + x) − x} + (

e−x − e−2x)

exp{

R3(5) − R3(5 + x)}, if 2 < x .

As seen in Fig. 2b, the survival curves corresponding to the two age vectors cross eachother, which means that the result max{X − t | X > t} ≤st max{X − s | X > s} doesnot hold even though s �m t . ��Corollary 1 Consider a bivariate GMO random vector X with generators S1, S2, S3.If S1 and S2 have the same exponential distribution and S3 is DFR, then for a coherentstructure τ, s �m t implies

[τ(X − t) | X > t

] ≥st

[τ(X) − t1 + t2

2

∣∣∣∣ τ(X) >

t1 + t22

].

Proof Since t ≥m t̄ = (t1 + t2, t1 + t2)/2, by Theorems 2 and 3, we have

[τ(X − t) | X > t ] ≥st [τ(X − t̄ ) | X > t̄]. (8)

On the other hand, Xi may be easily verified to be right tail increasing in X2/ i , fori = 1, 2, whenever (X1, X2) has a GMO distribution. Consequently, by Theorem11.2.4 of Li et al. (2013), we have

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[τ(X − t̄) | X > t̄

] ≥st

[τ(X) − t1 + t2

2

∣∣∣∣ τ(X) >

t1 + t22

].

Upon combining these two inequalities, we immediately obtain the desired result. ��

Recall that a non-negative bivariate random vector X = (X1, X2) is said to bebivariate new worse than used (B-NWU) if X ≤st [(X1−t, X2 −t) | X1 > t, X2 > t]for all t ≥ 0. One may refer to Pellerey (2008) and Li and Pellerey (2011) for examplesof bivariate distributions with this B-NWU property.

Corollary 2 Suppose X has a bivariate GMO distribution with generators S1, S2, S3.If S1 and S2 have the same exponential distribution and S3 is DFR, then for a coherentstructure τ, [τ(X − t) | X > t] ≥st τ(X) for any t .

Proof According to Corollary 4.2 of Li and Pellerey (2011), X is B-NWU. Then, forany t ,

τ(X) ≤st

[τ(X) − t1 + t2

2| X1 >

t1 + t22

, X2 >t1 + t2

2

].

Upon combining this with the inequality in (8), the stochastic order τ(X) ≤st [τ(X−t)| X > t ] follows readily. ��

5 Minimum of residuals of lifetimes following TTE models

Assume a bivariate random vector X ∼ TTE(W, R) with

P(X1−t1 > x, X2 − t2 > y | X > t)=W (R(t1 + x)+R(t2 + y))/W (R(t1) + R(t2)).

If R(x) = λx for some λ > 0, then, for any x, y ≥ 0 and t ,

P(X1 − t1 > x, X2 − t2 > y | X > t)) = W (λ(t1 + t2 + x + y))/W (λ(t1 + t2))

is invariant with respect to t1 + t2. As a result, for any coherent system with structure

function τ , we always have [τ(X − t) | X > t] st= [τ(X − s) | X > s] whenevers �m t .

In this section, we establish some sufficient conditions for the majorization of theage vector of lifetimes following TTE models to imply the stochastic order of theminimum of residual lifetimes.

Theorem 4 For X ∼ TTE(W, R) with a log-convex W , if the hazard rate r(x) isincreasing (decreasing) and concave (convex), then s �m t implies

min{X − t | X > t} ≥st (≤st) min{X − s | X > s}.

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Proof Let us denote α(x, s) = R(x1 + s) + R(x2 + s) for any x and s ≥ 0. Since thehazard rate r(t) is increasing, the cumulative hazard R is convex. Note that t2 − s2 =s1 − t1, and so

R(t2 + x) − R(s2 + x) ≥ R(s1 + x) − R(t1 + x), for x ≥ 0.

This is just α(t, x) ≥ α(s, x) for x ≥ 0. Due to the increasing property of R, we alsohave α(s, x) ≥ α(s, 0) for x ≥ 0. Note that the log-convex property of W implies thatW (t + s)/W (t) is increasing in t for s ≥ 0. We thus have, for x ≥ 0,

W (α(t, x))

W (α(s, x))= W (α(s, x) + α(t, x) − α(s, x))

W (α(s, x))≥ W (α(s, 0) + α(t, x) − α(s, x))

W (α(s, 0)).

(9)On the other hand, the concavity of r(t) implies that R(t + x) − R(t) is concave in tfor any x ≥ 0. Since t2 − s2 = s1 − t1, we have, for any x ≥ 0,

[R(t2 + x) − R(t2)] − [R(s2 + x) − R(s2)]≤ [R(s1 + x) − R(s1)] − [R(t1 + x) − R(t1)],

or equivalently,

R(t1 + x) + R(t2 + x) − R(s1 + x) − R(s2 + x) + R(s1) + R(s2) ≤ R(t1) + R(t2);that is, α(t, x) − α(s, x) + α(s, 0) ≤ α(t, 0) for any x ≥ 0. Since W decreases, wehave

W (α(t, x) − α(s, x) + α(s, 0)) ≥ W (α(t, 0)), for x ≥ 0. (10)

Upon combining (9) and (10), we get W (α(t,x))W (α(s,x))

≥ W (α(t,0))W (α(s,0))

for x ≥ 0, i.e.,

W (α(t, x))

W (α(t, 0))≥ W (α(s, x))

W (α(s, 0)), for x ≥ 0.

Now, since

P(min{X − t | X > t} > x) = W (R(t1 + x) + R(t2 + x))

W (R(t1) + R(t2))= W (α(t, x))

W (α(t, 0)), x ≥ 0,

we immediately obtain P(min{X − t | X > t} > x) ≥ P(min{X − s | X > s} > x)

for x ≥ 0.The other part of the result may be proved in an analogous manner. ��Example 4 below shows that the decreasing property of the hazard rate assumed in

Theorem 4 can not be dropped.

Example 4 (Clayton copula) Let W (x) = 11+x and the hazard rate be r(x) = x + 16√

x.

Then,

W (x + t)

W (x)= 1 + x

1 + x + t= 1 − t

1 + x + t

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Fig. 3 Survival curve of minimum of residuals at s = (6, 8) (solid) and t = (4, 10) (dashed)

is increasing in x ≥ 0 for t ≥ 0, i.e., W is log-convex. Evidently, r(x) is not decreasingalthough it is convex in x ∈ (0,+∞).

For s = (6, 8) �m (4, 10) = t , we have

P(min{X − t | X > t} > x) = W (R(4 + x) + R(10 + x))/W (R(4) + R(10)),

P(min{X − s | X > s} > x) = W (R(6 + x) + R(8 + x))/W (R(6) + R(8)).

As seen in Fig. 3, the survival curve corresponding to t lies below the one corre-sponding to s. So, the result min{X − t | X > t} ≥st min{X − s | X > s} does nothold. ��

6 Extremes of residuals of i.i.d. lifetimes

Since both GMO and TTE models include independence as a special case, we payhere special attention to independent and identically distributed (i.i.d.) lifetimes andpresent some sufficient conditions for the majorization of the age vector to imply thestochastic order of the extremes of residual lifetimes.

Theorem 5 For X1 and X2 i.i.d. with a concave (convex) hazard rate, s �m t implies

min{X − t | X > t} ≥st (≤st) min{X − s | X > s}.

Proof Let r(x) and R(x) be the hazard rate and cumulative hazard of X1, respectively.The concave property of r(t) implies that R(t + x) − R(t) is concave for any x ≥ 0,i.e.,

[R(t2) − R(t2 + x)] − [R(t1 + x) − R(t1)]≥ [R(s2) − R(s2 + x)] − [R(s1 + x) − R(s1)].

Since X1 and X2 are i.i.d., we have, for x ≥ 0,

P(min{X − t | X > t} > x) = exp{R(t2) − R(t2 + x) + R(t1) − R(t1 + x)}

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Generalized Marshall–Olkin and time transformed exponential models

≥ exp{R(s2) − R(s2 + x) + R(s1) − R(s1 + x)}= P(min{X − s > x | X > s});

that is, min{X − t | X > t} ≥st min{X − s | X > s}.The reverse inequality for the result may be obtained in an analogous manner. ��

Theorem 6 For X1 and X2 i.i.d. with a monotone and concave hazard rate, s �m timplies

max{X − t | X > t} ≥st max{X − s | X > s}.

Proof Let r(x) and R(x) be the hazard rate and cumulative hazard of X1, respectively.Denote Hx (t) = 1 − exp{R(t) − R(t + x)} for any x ≥ 0. Since X1 and X2 are i.i.d.,we have, for x ≥ 0,

P(max{X − t | X > t} ≤ x)

= P(t1 ≤ X1 ≤ t1 + x, t2 ≤ X2 ≤ t2 + x)/P(t1 ≤ X1, t2 ≤ X2)

= P(t1 ≤ X1 ≤ t1 + x) · P(t2 ≤ X2 ≤ t2 + x)/P(t1 ≤ X1) · P(t2 ≤ X2)

= (1 − exp{R(t1) − R(t1 + x)}) · (

1 − exp{R(t2) − R(t2 + x)})= Hx (t1)Hx (t2).

If Hx (t) is log-concave in t , then Hx (t + s)/Hx (t) is decreasing in t for s ≥ 0. Witht2 − s2 = s1 − t1, it then follows that, for t1 ≤ s1 ≤ s2 ≤ t2,

Hx (t2)

Hx (s2)= Hx (s2 + t2 − s2)

Hx (s2)≤ Hx (t1 + s1 − t1)

Hx (t1)= Hx (s1)

Hx (t1),

and consequently Hx (t1)Hx (t2) ≤ Hx (s1)Hx (s2) for x ≥ 0; that is,

P(max{X − t | X > t} ≤ x) ≤ P(max{X − s | X > s} ≤ x).

This yields the desired result under log-concavity of Hx (t). Consequently, it sufficesto prove that Hx (t) is log-concave in t for any x ≥ 0 under the condition that thehazard rate r is monotone and concave.

Assume that r is increasing. Then, r(t + x) ≥ r(t) for x ≥ 0 and R is convex. Notethat

∂ log Hx (t)

∂t= − exp{R(t) − R(t + x)}

1 − exp{R(t) − R(t + x)} [r(t) − r(t + x)]

=( 1

1 − exp{R(t) − R(t + x)} − 1)[r(t + x) − r(t)]

= ([Hx (t)]−1 − 1)[r(t + x) − r(t)], (11)

where the reciprocal [Hx (t)]−1 is decreasing in t for x ≥ 0. Due to the concavityof r, r(t + x) − r(t) is non-negative and decreasing in t for x ≥ 0. Thus, Hx (t) =

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Fig. 4 Distribution of maximum of residuals at s = (1, 1.5) (solid) and t = (.5, 2) (dashed)

1−exp{R(t)− R(t + x)} ∈ (0, 1] for x ≥ 0, implying [Hx (t)]−1 −1 ≥ 0. From (11),we can conclude that, for any x ≥ 0,

∂ log Hx (t)∂t is decreasing in t and so log Hx (t) is

concave in t .Assume that r is decreasing. Then, r(t + x) ≤ r(t) for x ≥ 0 and R is concave and

so R(t) − R(t + x) is increasing in t for x ≥ 0, and this implies that [Hx (t)]−1 − 1is increasing in t for x ≥ 0. The concave property of r implies r(t) − r(t + x) isnon-negative and increasing in t for x ≥ 0. Noting that [Hx (t)]−1 − 1 ≥ 0, it followsfrom (11) that, for x ≥ 0,

−∂ log Hx (t)

∂t=

([Hx (t)]−1 − 1

)[r(t) − r(t + x)]

is increasing in t , i.e., log Hx (t) is concave in t for any x ≥ 0. This completes theproof of the theorem. ��

The next example shows that the concavity of the hazard rate is necessary in The-orem 6.

Example 5 Assume X1 and X2 are i.i.d. with the hazard rate

r(x) ={

1/√

x, if 0 < x < 4,

1/(x − 2), if x ≥ 4.

Evidently, r(x) is decreasing in x ∈ (0,+∞). Note that r(x) is convex in x ∈ (0, 4),which means that r(x) is not concave. Set s = (1, 1.5) �m (0.5, 2) = t . We then have

P(max{X − t | X > t} > x)

= (1 − exp

{R(0.5) − R(0.5 + x)}) (

1 − exp{

R(2) − R(2 + x)}) ,

P(max{X − s | X > s} > x)

= (1 − exp

{R(1) − R(1 + x)}) (

1 − exp{

R(1.5) − R(1.5 + x)}) .

As seen in Fig. 4, the distributional curves corresponding to the two age vectorscross each other, which invalidates the ordering result max{X−t|X > t} ≥st max{X−s|X > s}. ��

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Generalized Marshall–Olkin and time transformed exponential models

Corollary 3 Suppose X1 and X2 are independent and identically distributed. If thehazard rate r(t) is monotone and concave, then, for a coherent structure τ ,

[τ(X − t) | X > t] ≥st

[τ(X) − t1 + t2

2

∣∣∣∣ τ(X) >t1 + t2

2

].

Proof Since t �m t̄ = (t1 + t2, t1 + t2)/2, by Theorems 5 and 6, we have

[τ(X − t) | X > t ] ≥st [τ(X − t̄ ) | X > t̄ ].

On the other hand, the independence implies the right tail increasing property. Con-sequently, by Theorem 11.2.4 of Li et al. (2013), we have

[τ(X − t̄) | X > t̄

] ≥st

[τ(X) − t1 + t2

2

∣∣∣∣ τ(X) >

t1 + t22

].

Upon combining the above two inequalities, we immediately obtain the desired result.��

7 Concluding remarks

We have discussed here as to how the heterogeneity of the age vector of two exchange-able used components has an impact on the resulting extremes in terms of stochasticorder. In this connection, it is useful to make the following closing remarks:

(i) In practice, extremes of more than two random risks or lifetimes will often beencountered. So, further research work may be done with either maximum orminimum of multiple random residual lifetimes. Work in this direction is expectedto be of interest in insurance industry.

(ii) Since order statistics, which correspond to k-out-of-n structures in reliability,include the maximum and the minimum as the two extremal cases, it will natu-rally be of interest to study the heterogeneity of the age vector in the order statis-tics of some residual lifetimes. This investigation will cast some new insight onscheduling the preventive inspection time points in industrial engineering.

(iii) The present work has focused on generalized Marshall–Olkin structure andArchimedean copula of the associated random lifetimes; however, in practice,it will be usually difficult to tell whether one of these two structures is applicableor not. A more practical situation will be to confine our attention to some non-parametric dependence structures, such as stochastic increasing and arrangementincreasing structures, and then to study the heterogeneity of the age vector insuch as nonparametric dependence setup.

Acknowledgments Authors thank two anonymous reviewers for their comments on an earlier version ofthe manuscript which resulted in this improved version.

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