Dynamic Multivariate Quantile Residual Life in Reliability Theory

Copyright © 2018 M. Shafaei Noughabi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We extend the univariate α-quantile residual life function tomultivariate 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.


Introduction
For a random lifetime , the -quantile residual life (-QRL) function proposed by Haines and Singpurwalla (1974) describes the -quantile of the well-known remaining lifetime of  given its survival at time  > 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 -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 called dynamic in the sense that it is a measure conditioned on an observed history (which can consist of some failures) up to time .
Recently, Shafaei Noughabi and Kayid [16] proposed a bivariate -QRL (-BQRL) function which characterizes the underlying distribution properly.Although this function is useful and applicable in statistics and reliability fields, it is nondynamic.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 , he/she may observe that (i) all belts may be safely working or (ii) one or more of them may fail at .The bivariate -QRL (-BQRL) function introduced by Shafaei Noughabi and Kayid [16] does not support histories of type (ii).This motivates us to extend the univariate -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 , the dynamic multivariate -QRL (-MQRL) function measures the -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 dynamic -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 -MQRL function.Also, a notion of positive dependency has been proposed and discussed.Section 5 investigates conditions defining the class of distributions with decreasing -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.

Preliminaries
Let random lifetime  be distributed on [0, ∞) according to continuous distribution .Then, the well-known hazard rate and -QRL functions are given, respectively, by and in which () = 1 − () shows the reliability function,  = 1 − , and  −1 () = inf{; () = } is the inverse function of .These two functions are related in the way of the simple equation which directly can be translated to It implies immediately that    () ≥ −1.Moreover, when the hazard rate is increasing (decreasing) at all points of the support, the -QRL exhibits a decreasing (increasing) form.
Applying straightforward algebra, it can be written as in which  −1  (; x (−) ) = inf{  : (x) = } and vector x (−) has dimension  − 1 and is obtained by removing the th element of x.This version of -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 of -MQRL with the multivariate hazard rate function.The proof is straightforward and hence omitted (cf.Shafaei Noughabi and Kayid [16]).
and in turn

Dynamic Multivariate 𝛼-Quantile Residual Life
Let  represent the reliability function of a bivariate random variable X.For brief representation, denote lim . The conditional hazard rate functions of X are (cf.Shaked and Shanthikumar [15] or Cox [17]) and Intuitively   (),  = 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  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 -BQRL functions by and where and Simple calculations imply and where and the expressions for  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 .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  into account.The next result provides the relation between conditional bivariate hazard rate functions and dynamic -BQRL.
Theorem 2. Let q, ( 1 ,  2 ),  = 1,2, have continuous differentiation functions with respect to their both coordinates.Then, we have and Proof.Due to the relation  ,1 () = q,1 (, ), the differentiation of  ,1 () can be described as the sum of differentiations in two directions Taking  = 2 and applying (10) with  = 1 gives the first expression.By some algebra for the second differentiation, we have which shows (21).Analogous statements indicate (22).To justify (23), we consider By differentiation from  with respect to  in the equation inside brackets and applying definition of  ,1 ( |  2 ) the result follows.In a similar way, (24) is obtained, and this complete the proof.
Next, we generalize the concept to multivariate setting.Let the nonnegative random vector X = ( 1 ,  2 , . . .,   ) accommodate distribution , and ℎ  captures history of events related to  components up to time , i.e., in which  = { 1 ,  2 , . . .,   } shows indices of events up to ,   is the complement of  with respect to  = {1, 2, . . ., }, and 1 is a vector of 1's with proper dimension.Note that 1 is the multiplication of scalar  by a vector 1 (a vector with same elements 1 and proper dimension) which clearly reduces to a vector with the same elements  and dimension of 1. Fix the history ℎ  as above, and then for any component  ∈   , the conditional hazard rate function can be written as which describes the probability of instant failure of component  at time , given history ℎ  .For empty set , we have initial hazard functions.Denote -quantile of a random variable  with reliability function  by   () =  −1 ().
Then for  ∈   , we define the -MQRL function at time  by which can be simplified to For a simple representation denote lim where which proves (39).To show (40), we can differentiate from both sides of with respect to , and hence the proof is completed.

Multivariate 𝛼-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  and  with reliability functions  and  in the univariate context.Statisticians apply different ordering criteria in their investigations.As a simple one,  is said to be smaller than  in the usual stochastic order,  ≤  , if for every  > 0, () ≤ ().In the multivariate framework, X ≤  Y when (X) ≤ (Y) for all nondecreasing functions  :  + →  + for which these expectations exist.As a stronger ordering,  is said to be smaller than  in the hazard rate order,  ≤ ℎ , if for all  1 ≤  2 .Provided that  and  are equipped with the hazard rate functions   and   respectively, condition (46) is equivalent with the inequality   () ≥   () for every  ≥ 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 time , the history ℎ  is referred to be more severe than ℎ  , denoted as ℎ  ⪯ ℎ  , if (i) every failed component in ℎ  also be failed in ℎ  ; (ii) for common failures in both ℎ  and ℎ  , the failures in ℎ  are earlier than failures in ℎ  .
More formally, and where  ∩  = 0, 01 ≤ x  ≤ y  ≤ 1 and 01 ≤ x  ≤ 1.In view of these notations, X is defined to be smaller than Y in the hazard rate order, X ≤ ℎ Y, if for every  ≥ 0 whenever ℎ   ⪯ ℎ   ,  ∈ ( ∪ )  ; that is,  shows a component survived in both histories and    and    are the multivariate hazard rate functions defined in (34) corresponding to X and Y, respectively.This order is not reflexive; that is, X ≤ ℎ X 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 of   's are identically zero.Without loss of generality, X can be rearranged such that just   ,  = 1, 2, . . ., , is identically zero and the rest of the vectors are absolutely continuous.Then, we say X ≤ ℎ X if (49) is true for  > .
As a weaker order,  is said to be smaller than  in the quantile residual life order, denoted as  ≤ − , if for every  ≥ 0 Franco-Pereira et.al. [22] proved that  ≤ ℎ  if for any  ∈ (0, 1) we have  ≤ − .Here, we define X to be smaller than Y in the -quantile residual life if order X ≤ − Y, if whenever ℎ   ⪯ ℎ   and for all components  alive in both histories.Like multivariate hazard rate order, it is not reflexive too.In fact X ≤ − X shows a positive dependency between components of X. Situations in which some of   's are identically zero can be treated as explained for multivariate hazard rate order.Theorem 5.For two vectors X and Y, X ≤ ℎ Y if and only if for every  ∈ (0, 1) we have X ≤ − Y.
Proof.Firstly, we show that X ≤ ℎ Y if and only if for every  ≥ 0,  ≥ 0, ℎ   ⪯ ℎ   , and  are alive in both of them To achieve this, let ℎ which is equivalent to the statement that for every  ≥ 0,  ≥ 0, which is apparently equivalent to (52).Thus, the result follows by definition of -MQRL function in (35) and (52), and the proof is completed.
Remark 6.Let  and two histories ℎ  ⪯ ℎ  be fixed.It can be seen from Shaked and Shanthikumar [15] that X ≤ ℎ Y implies in which (X − 1) + is the vector (( 1 − ) + , . . ., (  − ) + ) and (  − ) + = max{  − , 0},  = 1, . . ., .As can be seen from the proof of Theorem 5, we can write for every  ≥ 0 and ℎ  ⪯ ℎ  . (57) Assume that the structure of a lifetime vector X satisfies the following rule.For an alive component, the more severe history it belongs to, the smaller -MQRL values are expected; i.e., the -MQRL is decreasing in history ℎ  .Intuitively, this structure describes a positive dependency between lifetimes.More precisely, we say that X is -QRL decreasing upon failure (-QRL-DF) if for every  ≥ 0 where ℎ  ⪯ ℎ  and  is an alive component at time  in both histories.Apparently, it is equivalent to say that 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 -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 for X to be -QRL-DF is that for every  ⊂ {1, . . ., },  ∈   ,  ≥ 0, and 01 ≤ x  ≤ 1.
Proof.First, we state (60) in terms of -MQRL of X. Rename vectors of the left hand side and right hand side of (60) by X 1 and X 2 , respectively.Let , , and  be fixed and note that X 1 has one zero value and therefore its dimension is one unit less than dimension of X 2 .Then, (60) is equivalent to for every  ≥ 0, ℎ *  ⪯ ℎ *  and  ∈   − {} which is alive in both histories.Clearly, (58) implies (64) which by the fact that , , and  are arbitrary proves the necessity part.
To show sufficiency, assume that (60) holds and ℎ  ⪯ ℎ  are two arbitrary histories up to time  ≥ 0. We need to show that for every component  survived at  in ℎ  .Note that ℎ  requires at least one recorded failure with failure time strictly greater than maximum failure times of ℎ  , to be more sever than ℎ  .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  and x  , respectively.If there are not such common failures, set  = 0.Then, if  ̸ = 0, let  be the first component (which its existence is guaranteed) failed after components  and  denote its failure time.When  = 0, let  and  be the first failed component of ℎ  and its failure time, respectively.Now, two different cases are possible in history ℎ  about component : (i) it be alive at time  (ii) it fails at a time Based on these arrangements, ℎ  and ℎ  can be written in one of the following cases (a) or (b): while  ⊂ {1, . . ., },  ∈   ,  ≥ 0, and 01 ≤ x  ≤ 1 (cf.Shaked and Shanthikumar [11] for more information).Now by Theorem 5, it is immediate that X is SL ⇐⇒ X is -QRL-DF for all  ∈ (0, 1) . (67) Applying Theorem 1 from Shaked and Shanthikumar [12], this model follows MIFR property and in turn D -MQRL for every  ∈ (0, 1) if for every   ⊂   ⊂ {1, 2, . . ., }, where they show alive components at  and  ∈   .

Conclusion
The dynamic -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 -QRL of a survived component at time  > 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 -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 -QRL functions, as results show that similar spirit holds in the multivariate case.A new multivariate stochastic order, namely, -MQRL order, has been defined.The results reveal that -MQRL order is weaker than the multivariate hazard rate order.However, the -MQRL order for every  ∈ (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].(ii) Provide a nonparametric estimator of the -MQRL functions when X  ,  = 1, 2, . . ., , is a sample of  independent and identically distributed random vectors.
[23]ple 12. Ross[23]considered a system composed of  possibly dependent components, labelled by 1, 2, . . ., , 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,  ≥ 0, just depends on the set of alive components at that time.Suppose that   ⊂ {1, 2, . . ., } denote the set of alive components at some time .Then, for a component  ∈   , the failure rate function at ,   ( | ℎ  ), reduces to The class of multivariate distributions with decreasing -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 the -MQRL functions characterize the corresponding distributions.Specially, in the bivariate case it leads us to solving the following functional equations in terms of :  ( 1 () , ) =  (, ) ,  (,  2 ()) =  (, ) ,   () =  +  , (),  = 1, 2,  3 ( |  2 ) =  +  ,1 ( |  2 ), and  4 ( |  1 ) =  +  ,2 ( |  1 ).