# Dynamic Multivariate Quantile Residual Life in Reliability Theory

Authors: M. Shafaei Noughabi; M. Kayid; A. M. Abouammoh
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

We extend the univariateα-quantile residual life function to multivariate setting preserving its dynamic feature. Principal attributes of this function are derived and their relationship to the dynamic multivariate hazard rate function is discussed. A corresponding ordering, namely, α-quantile residual life order, for random vectors of lifetimes is introduced and studied. Based on the proposed ordering, a notion of positive dependency is presented. Finally, a discussion about conditions characterizing the class of decreasing multivariate α-quantile residual life functions is pointed out.


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## Body

## 1. Introduction

For a random lifetimeX, the α-quantile residual life (α-QRL) function proposed by Haines and Singpurwalla (1974) describes the α-quantile of the well-known remaining lifetime of X given its survival at time x>0. This function has been regarded as a prominent tool in reliability theory and survival analysis specially due to its potential advantages rather than the popular mean residual lifetime (MRL) function. Schmittlein and Morrison [1] discussed some of these advantages and applications of the median residual life function. Joe and Proschan [2], Gupta and Longford [3], Franco-Pereira and Uña-Álvarez [4], and Lillo [5] are among many authors who conducted their researches on the α-QRL function. Intuitively, we may deal with vectors of possibly dependent random lifetimes. In such situations, extending concerned concepts to multivariate setting allows us to treat the problems in the right way. The multivariate statistical methods play a crucial role in studying a wide variety of several complex engineering models. From many researchers who have extensively studied the multivariate lifetime measures, we refer to Johnson and Kotz [6], Arjas and Norros [7], Arnold and Zahedi [8], Baccelli and Makowski [9], Nair and Nair [10], Shaked and Shanthikumar [11, 12], Kulkarni and Rattihalli [13], and Hu et. al. [14]. Shaked and Shanthikumar [15] introduced and studied a dynamic version of the multivariate MRL function. This function is calleddynamic in the sense that it is a measure conditioned on an observed history (which can consist of some failures) up to time x.Recently, Shafaei Noughabi and Kayid [16] proposed a bivariate α-QRL (α-BQRL) function which characterizes the underlying distribution properly. Although this function is useful and applicable in statistics and reliability fields, it isnondynamic. In the areas of reliability theory, the dynamic residual life function authorizes engineers to track reliability of their systems at any time given any observed history. As an example, consider a machine having some belts working simultaneously. One engineer that tracks the machine may observe different types of histories. At arbitrary time x, he/she may observe that (i) all belts may be safely working or (ii) one or more of them may fail at x. The bivariate α-QRL (α-BQRL) function introduced by Shafaei Noughabi and Kayid [16] does not support histories of type (ii). This motivates us to extend the univariate α-quantile residual life function to multivariate setting preserving its dynamic feature. Now we are motivated to propose a dynamic measure which enables engineer to describe the belts lifetimes after observing any history, type (i) or (ii). For the components or subsystems survived until time x, the dynamic multivariate α-QRL (α-MQRL) function measures the α-quantile of the remaining lifetime conditioned on any possible history at this time. It can be regarded as a serious competitor for the multivariate MRL recommended by Shaked and Shanthikumar [15] and may even be preferred to that due to the comments of Schmittlein and Morrison [1].The rest of the paper is arranged as follows. The next section provides some preparative material that we need to develop the results. We start our results with the dynamicα-BQRL function and its basic behaviour in Section 3. In that section, the concept has been generalized to multivariate setting. Section 4 deals with a new stochastic order for random vectors based on the proposed α-MQRL function. Also, a notion of positive dependency has been proposed and discussed. Section 5 investigates conditions defining the class of distributions with decreasing α-MQRL functions and provides some related results. Finally, in Section 6, we give a brief conclusion and some remarks of the current and future of this research.Throughout the paper, we assume that the random vectorsX and Y follow absolutely continuous distributions on the support [0,∞)n. Moreover, to distinguish nondynamic multivariate functions from dynamic we insert a tilde (~) sign for nondynamic ones. Also, to provide succinct notations, denote (x1,…,xi-1,t,xi+1,…,xn) by xi;t.


## 2. Preliminaries

Let random lifetimeX be distributed on [0,∞) according to continuous distribution F. Then, the well-known hazard rate and α-QRL functions are given, respectively, by(1)λx=-ddxln⁡F-x,x≥0,and(2)qαx=F--1α-F-x-x,x≥0,in which F-(x)=1-F(x) shows the reliability function, α-=1-α, and F--1(p)=inf⁡{x;F-(x)=p} is the inverse function of F-. These two functions are related in the way of the simple equation(3)∫xx+qαxλxdx=-ln⁡α-,which directly can be translated to(4)1+qα′x=λxλx+qαx.It implies immediately that qα′(x)≥-1. Moreover, when the hazard rate is increasing (decreasing) at all points of the support, the α-QRL exhibits a decreasing (increasing) form.LetX=(X1,X2,…,Xn) follow the absolutely continuous reliability function F-. Johnson and Kotz [6] defined the multivariate hazard rate, as a vector(5)Λ~x=-∂∂x1,∂∂x2,…,∂∂xnln⁡F-x,x∈R+n.Denote the ith element of this vector by λ~i(x). We generalize the α-BQRL function proposed by Shafaei Noughabi and Kayid [16] to define the α-MQRL given by(6)q~αx=q~α,1x,q~α,2x,…,q~α,nx,x∈R+n,where(7)q~α,ix=inf⁡t:F-it;x=α-,with F-i(t;x)=P(Xi-xi>t∣X1>x1,…,Xn>xn). Applying straightforward algebra, it can be written as(8)q~α,ix=F-i-1α-F-x;x-i-xi,in whichF-i-1(p;x-i)=inf⁡{xi:F-(x)=p} and vector x(-i) has dimension n-1 and is obtained by removing the ith element of x. This version of α-MQRL gives a measure just for histories without experiencing any failure which violates its dynamicity. Nevertheless, it is sufficiently useful and applicable in reliability engineering and survival analysis to be studied in detail.The next result investigates the relation ofα-MQRL with the multivariate hazard rate function. The proof is straightforward and hence omitted (cf. Shafaei Noughabi and Kayid [16]).Lemma 1.
Consider the reliability functionF- with the hazard rate function Λ~(x)and the α-MQRL function q~α(x); then for every i=1,2,…,n we have(9)∫xixi+q~α,1xλ~ixi;tdt=-ln⁡α-,and in turn(10)1+∂∂xiq~α,ix=λ~ixλ~ixi;xi+q~α,ix.


## 3. Dynamic Multivariateα-Quantile Residual Life

LetF- represent the reliability function of a bivariate random variable X. For brief representation, denote limδ→0⁡1/δP(x1<X1≤x1+δ,X2>x2) by Pδ1(X1=x1,X2>x2) and similarly limδ→0⁡1/δP(X1>x1,x2<X2≤x2+δ) by Pδ2(X1>x1,X2=x2). The conditional hazard rate functions of X are (cf. Shaked and Shanthikumar [15] or Cox [17])(11)λix=limδ→0⁡1δPx<Xi≤x+δ∣X>x1,i=1,2,x≥0,λ1x∣x2=limδ→0⁡1δPx<X1≤x+δ∣X1>x,X2=x2,x≥x2≥0,and(12)λ2x∣x1=limδ→0⁡1δPx<X2≤x+δ∣X1=x1,X2>x,x≥x1≥0.Intuitivelyλi(x),  i=1,2, are referred to initial hazard rate functions in the sense that they measure the hazard rate for components before any failure. The underlying distribution F can be characterized uniquely by these four functions (cf. Cox [17]). Shaked and Shanthikumar [15] applied the conditional hazard rate functions in description attributes of dynamic bivariate MRL. We define the dynamic α-BQRL functions by(13)qα,ix=inf⁡t:F-it;x=α-,i=1,2,x≥0,qα,1x∣x2=inf⁡t:F-1∗t;x,x2=α-,x≥x2≥0,and(14)qα,2x∣x1=inf⁡t:F-2∗t;x1,x=α-,x≥x1≥0,where(15)F-it;x=PXi-x>t∣X1>x,X2>x,F-1∗t;x,x2=PX1-x>t∣X1>x,X2=x2,and(16)F-2∗t;x1,x=PX2-x>t∣X1=x1,X2>x.Simple calculations imply(17)qα,ix=F-i-1α-F-x,x;x-x,i=1,2,(18)qα,1x∣x2=F-1∗-1α-Pδ2X1>x,X2=x2;x2-x,and(19)qα,2x∣x1=F-2∗-1α-Pδ1X1=x1,X2>x;x1-x,where(20)F-1-1p;x=inf⁡t:F-t,x=p,F-1∗-1p;x2=inf⁡y:Pδ2X1>y,X2=x2=p,and the expressions for F-2-1(p;x) and F-2∗-1(p;x1) are analogous.Relations (17) to (19) give us the possibility of computing the quantiles of remaining life of the surviving components conditioning on the observed history up to time x. Thus, these functions may be relevant for engineers that deal with systems of multiple dependent objects. They can measure the remaining quantiles of surviving objects taking the effect of the observed history at any time x into account. The next result provides the relation between conditional bivariate hazard rate functions and dynamic α-BQRL.Theorem 2.
Letq~α,i(x1,x2),  i=1,2, have continuous differentiation functions with respect to their both coordinates. Then, we have(21)1+qα,1′x=λ~1x,xλ~1x+qα,1x,x+Pδ2X1>x+qα,1x∣x,X2=x-Pδ2X1>x+qα,1x,X2=xPδ1X1=x+qα,1x,X2>x,(22)1+qα,2′x=λ~2x,xλ~2x,x+qα,2x+Pδ1X1=x,X2>x+qα,2x∣x-Pδ1X1=x,X2>x+qα,2xPδ2X1>x,X2=x+qα,2x,(23)1+qα,1′x∣x2=λ1x∣x2λ1x+qα,1x∣x2∣x2,and(24)1+qα,2′x∣x1=λ2x∣x1λ2x+qα,2x∣x1∣x1.Proof.
Due to the relationqα,1(x)=q~α,1(x,x), the differentiation of qα,1(x) can be described as the sum of differentiations in two directions(25)∂∂xqα,1x=∂∂x1q~α,1x1,xx1=x+∂∂x2q~α,1x,x2x2=x,Taking n=2 and applying (10) with i=1 gives the first expression. By some algebra for the second differentiation, we have(26)1+qα,1′x=λ~1x,xλ~1x+qα,1x,x+Pδ2X1>x+qα,1x∣x,X2=x-Pδ2X1>x+qα,1x,X2=xPδ1X1=x+qα,1x,X2>x,which shows (21). Analogous statements indicate (22). To justify (23), we consider(27)qα,1x∣x2+x=inf⁡y:Pδ2X1>y,X2=x2=α-Pδ2X1>x,X2=x2.By differentiation from y with respect to x in the equation inside brackets and applying definition of qα,1(x∣x2) the result follows. In a similar way, (24) is obtained, and this complete the proof.Suppose thatX consists of two independent components X1 and X2. It is easily detectable that qα,1(x∣x2)=qα,1(x) and qα,2(x∣x1)=qα,2(x) for every x1 and x2, where qα,i(x) shows the α-QRL defined by (2) for Xi. In the sequel, we will see that a form of positive dependency between X1 and X2 implies qα,1(x)≥qα,1(x∣x2) and qα,2(x)≥qα,2(x∣x1). Therefore, for X1 and X2 positively dependent in such way, (21) and (22), respectively, imply(28)1+qα,1′x≥λ~1x,xλ~1x+qα,1x,x,and(29)1+qα,2′x≥λ~2x,xλ~2x,x+qα,2x.Fix x2 and suppose that λ1(x∣x2) is increasing in x and then (23) implies that qα,1(x∣x2) decreases in x. Similar argument holds for qα,2(x∣x1).Example 3.
The reliability function(30)F-x=exp⁡-θ1x1-θ2x2-θ12max⁡x1,x2,θ1,θ2,θ12≥0,x1,x2≥0,proposed by Marshal and Olkin [18] reveals a simple bivariate structure. Direct calculations show(31)qα,ix=-ln⁡α-θi+θ12,x≥0,i=1,2,qα,1x∣x2=-ln⁡α-θ1+θ12,x>x2≥0,and(32)qα,2x∣x1=-ln⁡α-θ2+θ12,x>x1≥0,exhibiting constant α-BQRL functions.Next, we generalize the concept to multivariate setting. Let the nonnegative random vectorX=(X1, X2,…,Xn) accommodate distribution F, and hx captures history of events related to n components up to time x, i.e.,(33)hx=XI=xI,XI′>x1,in which I={i1,i2,…,ik} shows indices of events up to x, I′ is the complement of I with respect to N={1,2,…,n}, and 1 is a vector of 1’s with proper dimension. Note that x1 is the multiplication of scalar x by a vector 1 (a vector with same elements 1 and proper dimension) which clearly reduces to a vector with the same elements x and dimension of 1. Fix the history hx as above, and then for any component j∈I′, the conditional hazard rate function can be written as(34)λjx∣hx=limδ→0⁡1δPx<Xj≤x+δ∣hx,x≥0,which describes the probability of instant failure of component j at time x, given history hx. For empty set I, we have initial hazard functions. Denote α-quantile of a random variable X with reliability function F- by Qα(X)=F--1(α-). Then for j∈I′, we define the α-MQRL function at time x by(35)qα,jx∣hx=QαXj-x∣hx,x≥0,which can be simplified to(36)qα,jx∣hx=inf⁡y:PXj-x>y∣XI=xI,XI′≥x1=α-=inf⁡y:PIXj>x+y,XI′≥x1,XI=xI=α-PIXI=xI,XI′≥x1.For a simple representation denote(37)limδi→0,i∈I⁡∏i∈I1δiPXI′>xI′,xi<Xi≤xi+δi,i∈Iby PδI(XI=xI,XI′>xI′). Observing history hx={X1>x,…,Xn>x} at x, i.e., I=∅, we have the initial α-MQRL functions that can be calculated as(38)qi,αx=inf⁡y:F-x,…,x+y︸ithelement,…,x=α-F-x,…,x=inf⁡y:F-x,…,x+y,…,x=α-F-x,…,x-x.Theorem 4.
Assume thatq~α,i(x1,…,xn),  i=1,2,…,n, defined by (7), have continuous differentiation functions with respect to their coordinates. If I=∅, we have a system of n equations(39)1+qα,i′x=λ~ix,…,xλ~ix,…,x+qα,ix,…,x+∑j=1,j≠inPδjXj=x,Xi>x+qα,ix∣hi,Xr>x;r=1,…n,r≠i,jPδiX1>x,…,Xi=x+qα,ix,…,Xn>x-PδjXj=x,Xi>x+qα,ix,Xr>x;r=1,…n,r≠i,jPδiX1>x,…,Xi=x+qα,ix,…,Xn>x,for i=1,…,n. Also, if we fix the history at x by hx={XJ=xJ,XJ′>x1}, where xJ<x1, then for any survived component i∈J′ at x we can write(40)1+qα,i′x∣hx=λix∣hxλi∗x+qα,ix∣hx∣chx+qα,iJ′-i+α-∑k∈J′≠inPδJ∪kXJ=xJ,Xk=x,XJ′-k>x1Pδi∪JXi=x+qα,ix∣hx,XJ′>x1,XJ=xJ-∑k∈J′≠inPδk∪JXJ=xJ,Xk=x,XJ′-k>x1,Xi>x+qα,ix∣hxPδi∪JXi=x+qα,ix∣hx,XJ′>x1,XJ=xJ,where(41)λi∗x+qα,ix∣hx∣chx+qα,iJ′-i=Pδi∪JXi=x+qα,ix∣hx,XJ′>x1,XJ=xJPδJXi>x+qα,ix∣hx,XJ′>x1,XJ=xJrepresents instantaneous risk of failure of component i by time x+qα,i(x∣hx), given history hx up to time x and censored history of components J′ except i after x.Proof.
As (38) shows, qα,i(x) equals q~α,i(x,…,x) defined by (7), so(42)qα,i′x=∑j=1n∂∂xjq~α,ix1j;xjxj=x.The ith term of this summation is(43)∂∂xiq~α,ix1i;xixi=x=λ~ix,…,xλ~ix,…,x+qα,ix,…,x-1.Other terms of it can be obtained by differentiation from both sides of the following equality with respect to xj at point x, for j=1,…,n,  j≠i:(44)F-x,…,x+qα,ix1j;xj,…,xnxj=α-F-x1j;xj,which proves (39). To show (40), we can differentiate from both sides of(45)PδJXi>x+qα,ix∣hx,XJ′>1x,XJ=xJ=α-PδJXJ=xJ,XJ′>1x,with respect to x, and hence the proof is completed.


## 4. Multivariateα-Quantile Residual Life Order

Stochastic orderings are very useful tools and have several applications in various areas such as probability, statistics, reliability engineering, and statistical decision theory. In literature, several concepts of stochastic orders between random variables have been given (cf. Shaked and Shanthikumar [19], as an excellent treatment of this topic). Consider two lifetime random variables X and Y with reliability functions F- and G- in the univariate context. Statisticians apply different ordering criteria in their investigations. As a simple one, X is said to be smaller than Y in the usual stochastic order, X≤stY, if for every x>0, F-(x)≤G-(x). In the multivariate framework, X≤stY when Eφ(X)≤Eφ(Y) for all nondecreasing functions φ:R+n→R+ for which these expectations exist.As a stronger ordering,X is said to be smaller than Y in the hazard rate order, X≤hrY, if(46)F-x2G-x1≤F-x1G-x2,for all x1≤x2. Provided that X and Y are equipped with the hazard rate functions λX and λY respectively, condition (46) is equivalent with the inequality λX(x)≥λY(x) for every x≥0.To extend some reliability and ageing concepts for random vectors, we should be able to compare possibly different histories. In the simplest case, we consider two histories with same lengths. At every timex, the history h-x is referred to be more severe than hx, denoted as hx⪯h-x, if(i)
every failed component inhx also be failed in h-x;(ii)
for common failures in bothhx and h-x, the failures in h-x are earlier than failures in hx.More formally,(47)h-xX=XI=xI,XJ=xJ,XI∪J′>x1,and(48)hxY=YI=yI,YI′>x1,where I∩J=∅,  01≤xI≤yI≤x1 and 01≤xJ≤x1. In view of these notations, X is defined to be smaller than Y in the hazard rate order, X≤hrY, if for every x≥0(49)λjXx∣h-xX≥λjYx∣hxY,whenever hxY⪯h-xX,  j∈(I∪J)′; that is, j shows a component survived in both histories and λjX and λjY are the multivariate hazard rate functions defined in (34) corresponding to X and Y, respectively. This order is not reflexive; that is, X≤hrX is not necessarily the case and implies a kind of positive dependency, namely,hazard increasing upon failure(cf. Shaked and Shanthikumar [11, 20] and Belzunce et. al. [21]).In many problems, we may deal with situations in which some ofXi’s are identically zero. Without loss of generality, X can be rearranged such that just Xi, i=1,2,…,k, is identically zero and the rest of the vectors are absolutely continuous. Then, we say X≤hrX if (49) is true for j>k.As a weaker order,X is said to be smaller than Y in the α-quantile residual life order, denoted as X≤α-qY, if for every x≥0(50)qαXx≤qαYx.Franco-Pereira et. al. [22] proved that X≤hrY if for any α∈(0,1) we have X≤α-qY. Here, we define X to be smaller than Y in the α-quantile residual life if order X≤α-qY, if(51)qα,jXx∣h-xX≤qα,jYx∣hxY,whenever hxY⪯h-xX and for all components j alive in both histories. Like multivariate hazard rate order, it is not reflexive too. In fact X≤α-qX shows a positive dependency between components of X. Situations in which some of Xi’s are identically zero can be treated as explained for multivariate hazard rate order.Theorem 5.
For two vectorsX and Y, X≤hrY if and only if for every α∈(0,1) we have X≤α-qY.Proof.
Firstly, we show thatX≤hrY if and only if for every x≥0,  t≥0,  hxY⪯h-xX, and j are alive in both of them(52)PXj-x>t∣h-xX≤PYj-x>t∣hxY.To achieve this, let h-xX={XI=xI,XJ=xJ,X(I∪J)′>x1} and hxY={YI=x~I,YI′>x1}. We notice that X≤hrY if(53)limϵ→0⁡1ϵPδIx<Yj<x+ϵ,hxYPδI∪Jh-xX-limϵ→0⁡1ϵPδI∪Jx<Xj<x+ϵ,h-xXPδIhxY≤0,which is equivalent to the statement that(54)PδI∪Jh-xXPδIhxYis decreasing in the jth survived component at x. Denote {YI=x~I,YI′>x1,Yj>y} by hxY[j,y] and assume similar notation h-xX[j,x]. Now it is easy to check that (54) is the case if(55)PδI∪Jh-xXj,x+tPδI∪Jh-xXj,x≤PδIhxYj,x+tPδIhxYj,x,for every x≥0,  t≥0, which is apparently equivalent to (52). Thus, the result follows by definition of α-MQRL function in (35) and (52), and the proof is completed.Remark 6.
Letx and two histories hx⪯h-x be fixed. It can be seen from Shaked and Shanthikumar [15] that X≤hrY implies(56)X-x1+∣h-x≤stY-x1+∣hx,in which (X-x1)+ is the vector ((X1-x)+,…,(Xn-x)+) and (Xi-x)+=max⁡{Xi-x,0}, i=1,…,n. As can be seen from the proof of Theorem 5, we can write(57)X≤hrY⇔X-x1+∣h-x≤stY-x1+∣hxfor  everyx≥0andhx⪯h-x.Assume that the structure of a lifetime vectorX satisfies the following rule. For an alive component, the more severe history it belongs to, the smaller α-MQRL values are expected; i.e., the α-MQRL is decreasing in history hx. Intuitively, this structure describes a positive dependency between lifetimes. More precisely, we say that X is α-QRL decreasing upon failure (α-QRL-DF) if for every x≥0(58)qα,iXx∣h-x≤qα,iXx∣hx,where hx⪯h-x and i is an alive component at time x in both histories. Apparently, it is equivalent to say that(59)X≤α-qX.Shaked and Shanthikumar [15] discussed a similar dependency based on the multivariate MRL, namely, MRL-DF property. The condition investigated in the following theorem provides a simpler investigation of α-QRL-DF property which is similar to characterizations for MRL-DF, weakened by failure (WBF), supportive lifetimes (SL), hazard rate increase upon failure (HIF), and multivariate totally positive of order 2 (MTP2) presented in Shaked and Shanthikumar [11, 15].Theorem 7.
A sufficient and necessary condition forX to be α-QRL-DF is that(60)XI′-x1∣XI=xI,Xj=x,XI′-j>x1≤α-qXI′-x1∣XI=xI,XI′>x1,for every I⊂{1,…,n},  j∈I′, x≥0, and 01≤xI≤x1.Proof.
First, we state (60) in terms of α-MQRL of X. Rename vectors of the left hand side and right hand side of (60) by X1 and X2, respectively. Let I, j, and x be fixed and note that X1 has one zero value and therefore its dimension is one unit less than dimension of X2. Then, (60) is equivalent to(61)qα,kX1y∣h-y∗≤qα,kX2y∣hy∗,whence hy∗⪯h-y∗; k∈I′-{j} shows a component alive at time y at both histories hy∗ and h-y∗ and y≥0. On the other hand, we have(62)qα,kX1y∣h-y∗=qα,kXx+y∣XI=xI,Xj=x,h-y∗,and(63)qα,kX2y∣hy∗=qα,kXx+y∣XI=xI,hy∗Thus, we can write (61) in the form(64)qα,kXx+y∣XI=xI,Xj=x,h-y∗≤qα,kXx+y∣XI=xI,hy∗,for every y≥0,  hy∗⪯h-y∗ and k∈I′-{i} which is alive in both histories. Clearly, (58) implies (64) which by the fact that I, j, and x are arbitrary proves the necessity part.
To show sufficiency, assume that (60) holds and ht⪯h-t are two arbitrary histories up to time t≥0. We need to show that(65)qα,kXt∣h-t≤qα,kXt∣htfor every component k survived at t in h-t. Note that h-t requires at least one recorded failure with failure time strictly greater than maximum failure times of ht, to be more sever than ht. If both of these histories contain same failed components which occur in same times, denote this set of common failed components and their failure times by I and xI, respectively. If there are not such common failures, set I=∅. Then, if I≠∅, let j be the first component (which its existence is guaranteed) failed after components I and x denote its failure time. When I=∅, let j and x be the first failed component of h-t and its failure time, respectively. Now, two different cases are possible in history ht about component j:(i)
it be alive at timet(ii)
it fails at a timex° such that x≤x°≤t.Based on these arrangements,h-t and ht can be written in one of the following cases (a) or (b):
(a)
h-t={XI=xI,Xj=x,XA=xA,XB=xB,XC>t1},andht={XI=xI,XA=x°A,XB∪C∪{j}>t1},wherex1≤xA≤x°A≤t1 and x1≤xB≤t1.(b)
h-t={XI=xI,Xj=x,XA=xA,XB=xB,XC>t1},andht={XI=xI,Xj=x°,XA=x°A,XB∪C>t1},wherex1≤xA≤x°A≤t1,  x1≤xB≤t1, and x≤x°≤t.
In either case, (60) implies that qα,kX(t∣h-t)≤qα,kX(t∣ht) for every alive component k at time t in history h-t, and this completes the proof.Remark 8.
A random vectorX is said to hold positive dependency property SL if(66)XI′-x1∣XI=xI,Xj=x,XI′-j>x1≤hrXI′-x1∣XI=xI,XI′>x1,while I⊂{1,…,n},  j∈I′,  x≥0, and 01≤xI≤x1 (cf. Shaked and Shanthikumar [11] for more information). Now by Theorem 5, it is immediate that(67)Xis  SL⇔Xisα-QRL-DF  for  allα∈0,1.


## 5. Decreasingα-MQRL Class of Life Distributions

The study of changes in the properties of any model, as the constituent components vary, is of great interest. A univariate random lifetimeX is said to accommodate decreasing α-QRL if qα(x) is decreasing in x≥0 or one of the following equivalent conditions hold:(68)X-x′∣X≥x′≤α-qX,(69)X-x′∣X≥x′≤α-qX-x∣X≥x,x′≥x≥0.To extend these conditions in the dynamic multivariate setting, we need to generalize the concept of comparing severity of two histories, discussed in Section 4, in the case that their lengths are not necessarily equal. Consider two histories hx and h-x′ with different lengths x≤x′. Then, h-x′ is referred to be more severe than hx, notationally hx⪯h-x′, if hx⪯h-x, where h-x contains all information of h-x′ over [0,x].Define a shift operatorθx on a random vector X by θxX=(X-x1)+,  x≥0. For a random vector X, we extend the arguments (68) and (69), respectively, by(70)θxX∣hx≤α-qX,x≥0,for every arbitrary history hx and(71)θx′′X∣h-x′′≤α-qθx′X∣hx′,x′′≥x′≥0,whence hx′⪯h-x′′. These extensions are similar to conditions for multivariate decreasing mean residual life by Shaked and Shanthikumar [15]. Another possible extension of (69) is(72)θx′X∣hx′≤α-qθxX∣hx,x′≥x≥0,where hx and hx′ coincide on the interval [0,x]. This means that all failed components in hx are failed in hx′ too with equal failure times. The condition (72) is similar to the multivariate increasing failure rate, proposed in Arjas [7], and a multivariate decreasing mean residual life notion of Shaked and Shanthikumar [15].Here we need some further notations to prove the following result. Fora<b, let h[a,b] represent a history which gives us those components alive at a and components failed in the interval [a,b] along with their failure times. Then, the previously defined history ht is same as h[0,t]. We say that h-[a+u,b+u], u≥0, is more severe than h[a,b], denoted as h[a,b]⪯h-[a+u,b+u], if every alive component at a+u in history h-[a+u,b+u] is alive at a in h[a,b] too and if a component fails in history h[a,b] at time x, it also fails in h-[a+u,b+u] at some time x0+u such that x0≤x.Let0≤a<b<c be fixed times. For two histories h[a,b], and h-[b,c], define the aggregated history h[a,b]⊕h-[b,c] to be a history over the interval [a,c], which shows the same alive components at time a as h[a,b] shows at a and contains all of the information that h[a,b] and h-[b,c] have, respectively, on the intervals [a,b] and [c,d].Theorem 9.
Three conditions (70), (71), and (72) are equivalent.Proof.
It is readily detectable that(73)71⇒72⇒70.Thus, it is sufficient to show that (70) ⇒ (71). Let (70) be true. Thus, we have(74)qα,jXx+s∣hx⊕h-x,x+s1≤qα,jXs∣h0,s1,for every x,s≥0,  h[0,s]1⪯h-[x,x+s]1, and hx, for components j alive at x+s in history h-[x,x+s]1. Now we must show that X satisfies (71), that is,(75)qα,jXx′′+t∣ħ-x′′⊕h-x′′,x′′+t2≤qα,jXx′+t∣ħx′⊕hx′,x′+t2,for every x′′≥x′≥0,  t≥0,  ħx′⪯ħ-x′′, and h[x′,x′+t]2⪯h-[x′′,x′′+t]2.
Lett, x′′, x′, ħx′⪯ħ-x′′, and h[x′,x′+t]2⪯h-[x′′,x′′+t]2 be given. We set s=x′+t≥0 and x=x′′-x′≥0. Moreover, set h[0,s]1, h-[x,x+s]1, and hx to be ħx′⊕h[x′,x′+t]2, ħ-[x′′-x′,x′′]⊕h-[x′′,x′′+t]2, and ħ-[0,x′′-x′]. By these considerations, it can be easily checked that (74) implies (75), which completes the proof immediately.In the presence of Theorem9, the following definition is well defined.Definition 10.
A random vectorX is decreasing multivariate α-quantile residual life (D α-MQRL) if one of the conditions (70), (71), or (72) is the case.
IfX satisfies condition (70), then apparently we have X≤α-qX. This means that the D α-MQRL implies the positive dependency property α-QRL-DF.
Shaked and Shanthikumar [12] proposed X to be multivariate increasing failure rate (MIFR) when(76)θx′X∣hx′≤hrX,for every x′≥0 and history hx′. It is a proper extension of the following condition:(77)X-x′∣X≥x′≤hrX.When it holds, X is said to accommodate an increasing failure rate form.Theorem 11.
The random vectorX is MIFR if it is D α-MQRL for every α∈(0,1).Proof.
The proof is followed immediately by (70), (76), and Theorem 5.Example 12.
Ross [23] considered a system composed of n possibly dependent components, labelled by 1,2,…,n, starting their work at time zero. The system satisfies a Markovian property in the sense that the failure rate of an alive component at any time, namely, t≥0, just depends on the set of alive components at that time. Suppose that I′⊂{1,2,…,n} denote the set of alive components at some time t. Then, for a component k∈I′, the failure rate function at t, λk(t∣ht), reduces to(78)λkt∣I′=limΔ→0⁡1ΔPt≤Xk<t+Δ∣I′.Applying Theorem 1 from Shaked and Shanthikumar [12], this model follows MIFR property and in turn D α-MQRL for every α∈(0,1) if(79)λkt∣J′≥λkt∣I′,for every J′⊂I′⊂{1,2,…,n}, where they show alive components at t and k∈J′.


## 6. Conclusion

The dynamicα-MQRL measure proposed in this paper is useful in both theoretical and applied aspects of reliability theory and survival analysis. It has been shown that this measure is closely related to the conditional hazard rate functions. In the bivariate case, the α-QRL of a survived component at time x>0, given that the other one has failed at some prior time, is decreasing when its corresponding conditional hazard rate is increasing. However, the behaviour of initial α-QRL functions is affected by the dependency structure of the components. When the corresponding conditional hazard rates are increasing, the positive dependency relieves the rate of descending of initial α-QRL functions, as results show that similar spirit holds in the multivariate case. A new multivariate stochastic order, namely, α-MQRL order, has been defined. The results reveal that α-MQRL order is weaker than the multivariate hazard rate order. However, the α-MQRL order for every α∈(0,1) implies the multivariate hazard rate order. Like some other multivariate orders, this order is not reflexive too. In fact, the statement that a vector is less than itself implies a positive dependency structure between components. This dependency is weaker than the supportive lifetime property discussed in Shaked and Shanthikumar [11]. The class of multivariate distributions with decreasing α-MQRL functions has been defined. It has been shown that this class includes the class of distributions following increasing multivariate hazard rate functions. Nevertheless, the following topics are interesting and still remain as open problems:(i)
Find out how closely theα-MQRL functions characterize the corresponding distributions. Specially, in the bivariate case it leads us to solving the following functional equations in terms of F-:(80)F-φ1x,x=α-F-x,x,F-x,φ2x=α-F-x,x,∂∂x2F-φ3x∣x2,x2=α-∂∂x2F-x,x2,∂∂x1F-x1,φ4x∣x1=α-∂∂x1F-x1,xwhere φi(x)=x+qα,i(x),  i=1,2,  φ3(x∣x2)=x+qα,1(x∣x2), and φ4(x∣x1)=x+qα,2(x∣x1).(ii)
Provide a nonparametric estimator of theα-MQRL functions when Xi,  i=1,2,…,n, is a sample of n independent and identically distributed random vectors.



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*Source: 1245656-2018-08-26.xml*