Abstract

We extend the univariate -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, -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 -quantile residual life functions is pointed out.

1. 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 . 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.

Throughout the paper, we assume that the random vectors and follow absolutely continuous distributions on the support . Moreover, to distinguish nondynamic multivariate functions from dynamic we insert a tilde () sign for nondynamic ones. Also, to provide succinct notations, denote by .

2. Preliminaries

Let random lifetime be distributed on according to continuous distribution . Then, the well-known hazard rate and -QRL functions are given, respectively, byandin which shows the reliability function, , and is the inverse function of . These two functions are related in the way of the simple equationwhich directly can be translated toIt implies immediately that . Moreover, when the hazard rate is increasing (decreasing) at all points of the support, the -QRL exhibits a decreasing (increasing) form.

Let follow the absolutely continuous reliability function . Johnson and Kotz [6] defined the multivariate hazard rate, as a vectorDenote the th element of this vector by . We generalize the -BQRL function proposed by Shafaei Noughabi and Kayid [16] to define the -MQRL given bywherewith . Applying straightforward algebra, it can be written as

in which and vector has dimension and is obtained by removing the th element of . 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]).

Lemma 1. Consider the reliability function with the hazard rate function and the -MQRL function ; then for every we haveand in turn

3. Dynamic Multivariate -Quantile Residual Life

Let represent the reliability function of a bivariate random variable . For brief representation, denote by and similarly by . The conditional hazard rate functions of are (cf. Shaked and Shanthikumar [15] or Cox [17])and

Intuitively ,  , 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 byandwhereandSimple calculations implyandwhereand the expressions for and 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 . 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 ,  , have continuous differentiation functions with respect to their both coordinates. Then, we haveand

Proof. Due to the relation , the differentiation of can be described as the sum of differentiations in two directionsTaking and applying (10) with gives the first expression. By some algebra for the second differentiation, we havewhich shows (21). Analogous statements indicate (22). To justify (23), we considerBy differentiation from with respect to in the equation inside brackets and applying definition of the result follows. In a similar way, (24) is obtained, and this complete the proof.

Suppose that consists of two independent components and It is easily detectable that and for every and , where shows the -QRL defined by (2) for . In the sequel, we will see that a form of positive dependency between and implies and . Therefore, for and positively dependent in such way, (21) and (22), respectively, implyandFix and suppose that is increasing in and then (23) implies that decreases in . Similar argument holds for .

Example 3. The reliability functionproposed by Marshal and Olkin [18] reveals a simple bivariate structure. Direct calculations showandexhibiting constant -BQRL functions.

Next, we generalize the concept to multivariate setting. Let the nonnegative random vector , accommodate distribution , and captures history of events related to components up to time , i.e.,in which shows indices of events up to , is the complement of with respect to , and is a vector of ’s with proper dimension. Note that is the multiplication of scalar by a vector (a vector with same elements 1 and proper dimension) which clearly reduces to a vector with the same elements and dimension of . Fix the history as above, and then for any component , the conditional hazard rate function can be written aswhich describes the probability of instant failure of component at time , given history . For empty set , we have initial hazard functions. Denote -quantile of a random variable with reliability function by . Then for , we define the -MQRL function at time bywhich can be simplified toFor a simple representation denoteby . Observing history at , i.e., , we have the initial -MQRL functions that can be calculated as

Theorem 4. Assume that ,  , defined by (7), have continuous differentiation functions with respect to their coordinates. If , we have a system of equationsfor . Also, if we fix the history at by , where , then for any survived component at we can writewhererepresents instantaneous risk of failure of component by time , given history up to time and censored history of components except after .

Proof. As (38) shows, equals defined by (7), soThe th term of this summation isOther terms of it can be obtained by differentiation from both sides of the following equality with respect to at point , for ,  :which proves (39). To show (40), we can differentiate from both sides ofwith respect to , and hence the proof is completed.

4. Multivariate -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 , . In the multivariate framework, when for all nondecreasing functions for which these expectations exist.

As a stronger ordering, is said to be smaller than in the hazard rate order, , iffor all . Provided that and are equipped with the hazard rate functions and respectively, condition (46) is equivalent with the inequality for every .

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 is referred to be more severe than , denoted as , if(i)every failed component in also be failed in ;(ii)for common failures in both and , the failures in are earlier than failures in .

More formally,andwhere ,   and . In view of these notations, is defined to be smaller than in the hazard rate order, , if for every whenever ,  ; that is, shows a component survived in both histories and and are the multivariate hazard rate functions defined in (34) corresponding to and , respectively. This order is not reflexive; that is, 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, can be rearranged such that just , , is identically zero and the rest of the vectors are absolutely continuous. Then, we say 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 Franco-Pereira et. al. [22] proved that if for any we have . Here, we define to be smaller than in the -quantile residual life if order , ifwhenever and for all components alive in both histories. Like multivariate hazard rate order, it is not reflexive too. In fact shows a positive dependency between components of . Situations in which some of ’s are identically zero can be treated as explained for multivariate hazard rate order.

Theorem 5. For two vectors and , if and only if for every we have .

Proof. Firstly, we show that if and only if for every ,  ,  , and are alive in both of themTo achieve this, let and . We notice that ifwhich is equivalent to the statement thatis decreasing in the th survived component at . Denote by and assume similar notation . Now it is easy to check that (54) is the case iffor every ,  , 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 be fixed. It can be seen from Shaked and Shanthikumar [15] that impliesin which is the vector and , . As can be seen from the proof of Theorem 5, we can write

Assume that the structure of a lifetime vector 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 . Intuitively, this structure describes a positive dependency between lifetimes. More precisely, we say that is -QRL decreasing upon failure (-QRL-DF) if for every where 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 to be -QRL-DF is thatfor every ,  , , and .

Proof. First, we state (60) in terms of -MQRL of . Rename vectors of the left hand side and right hand side of (60) by and , respectively. Let , , and be fixed and note that has one zero value and therefore its dimension is one unit less than dimension of . Then, (60) is equivalent towhence ; shows a component alive at time at both histories and and . On the other hand, we haveandThus, we can write (61) in the formfor every ,   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 are two arbitrary histories up to time . We need to show thatfor every component survived at in . Note that requires at least one recorded failure with failure time strictly greater than maximum failure times of , to be more sever than . 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 , respectively. If there are not such common failures, set . Then, if , let be the first component (which its existence is guaranteed) failed after components and denote its failure time. When , let and be the first failed component of and its failure time, respectively. Now, two different cases are possible in history about component :(i)it be alive at time (ii)it fails at a time such that .Based on these arrangements, and can be written in one of the following cases (a) or (b):(a),and,where and .(b),and,where ,  , and .In either case, (60) implies that for every alive component at time in history , and this completes the proof.

Remark 8. A random vector is said to hold positive dependency property SL ifwhile ,  ,  , and (cf. Shaked and Shanthikumar [11] for more information). Now by Theorem 5, it is immediate that

5. Decreasing -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 lifetime is said to accommodate decreasing -QRL if is decreasing in or one of the following equivalent conditions hold: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 and with different lengths . Then, is referred to be more severe than , notationally , if , where contains all information of over .

Define a shift operator on a random vector by ,  . For a random vector , we extend the arguments (68) and (69), respectively, byfor every arbitrary history andwhence . These extensions are similar to conditions for multivariate decreasing mean residual life by Shaked and Shanthikumar [15]. Another possible extension of (69) iswhere and coincide on the interval . This means that all failed components in are failed in 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. For , let represent a history which gives us those components alive at and components failed in the interval along with their failure times. Then, the previously defined history is same as . We say that , , is more severe than , denoted as , if every alive component at in history is alive at in too and if a component fails in history at time , it also fails in at some time such that .

Let be fixed times. For two histories , and , define the aggregated history to be a history over the interval , which shows the same alive components at time as shows at and contains all of the information that and have, respectively, on the intervals and .

Theorem 9. Three conditions (70), (71), and (72) are equivalent.

Proof. It is readily detectable thatThus, it is sufficient to show that (70) ⇒ (71). Let (70) be true. Thus, we havefor every ,  , and , for components alive at in history . Now we must show that satisfies (71), that is,for every ,  ,  , and .
Let , , , , and be given. We set and . Moreover, set , , and to be , , and . By these considerations, it can be easily checked that (74) implies (75), which completes the proof immediately.

In the presence of Theorem 9, the following definition is well defined.

Definition 10. A random vector is decreasing multivariate -quantile residual life (D -MQRL) if one of the conditions (70), (71), or (72) is the case.
If satisfies condition (70), then apparently we have . This means that the D -MQRL implies the positive dependency property -QRL-DF.
Shaked and Shanthikumar [12] proposed to be multivariate increasing failure rate (MIFR) whenfor every and history . It is a proper extension of the following condition:When it holds, is said to accommodate an increasing failure rate form.

Theorem 11. The random vector is MIFR if it is D -MQRL for every .

Proof. The proof is followed immediately by (70), (76), and Theorem 5.

Example 12. Ross [23] considered a system composed of possibly dependent components, labelled by , 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, , just depends on the set of alive components at that time. Suppose that denote the set of alive components at some time . Then, for a component , the failure rate function at , , reduces toApplying Theorem 1 from Shaked and Shanthikumar [12], this model follows MIFR property and in turn D -MQRL for every iffor every , where they show alive components at and .

6. 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 , 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 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 -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 :where ,  ,  , and .(ii)Provide a nonparametric estimator of the -MQRL functions when ,  , is a sample of independent and identically distributed random vectors.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All of the authors have equally made contributions. All authors read and approved the final manuscript.

Acknowledgments

The second and third authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for funding this research group (no. RGP-1435-036).