Introduction

For a random lifetime X, the ?-quantile residual life (?-QRL) function proposed by Haines and Singpurwalla (1974) describes the ?-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 (1981) discussed some of these advantages and applications of the median residual life function. Joe and Proschan (1984), Gupta and Longford (1984), Franco-Pereira and Uña-Álvarez (2013) and Lillo (2005) are among many authors which 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 (1975), Arjas and Norros (1984), Arnold and Zahedi (1988), Baccelli and Makowski (1989), Nair and Nair (1989), Shaked and Shanthikumar (1990, 1991a), Kulkarni and Rattihalli (2002) and Hu et. al. (2003). Shaked and Shanthikumar (1991b) 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 x.

Recently, Shafaei Noughabi and Kayid (2017) 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 non-dynamic. 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 ?-QRL (?-BQRL) function introduced by Shafaei Noughabi and Kayid (2017) 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 x, 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 (1991b) and may even be preferred to that due to the comments of Schmittlein and Morrison (1981).

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 behavior in Section <ref>96020601</ref>. In that section, the concept has been generalized to multivariate setting. Section <ref>96020603</ref> 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 <ref>96020605</ref> investigates conditions defining the class of distributions with decreasing ?-MQRL functions and provides some related results. Finally, in Section <ref>96020606</ref>, 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 X and Y follow absolutely continuous distributions on the support 0,?)?. Moreover, to distinguish non-dynamic multivariate functions from dynamic we insert a tilde (?) sign for non-dynamic ones. Also, to provide succinct notations, denote.

Preliminaries

Let random lifetime X be distributed on 0,?) according to continuous distribution F. Then, the well-known hazard rate and ?-QRL functions are given respectively by

andin which F(x)=1-F(x) shows the reliability function, ?=1-? 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 following simple equation

which directly can be translated to

It implies immediately that q_{?}?(x)?-1. Moreover, when the hazard rate is increasing (decreasing) at all points of the support, the ?-QRL exhibits a decreasing (increasing) form.

Let X=(X?,X?,…,X_{n}) follow the absolutely continuous reliability function F. Johnson and Kotz (1975) defined the multivariate hazard rate, as a vector

Denote the ith element of this vector by ?_{i}(x). We generalize the ?-BQRL function proposed by Shafaei Noughabi and Kayid (2017) to define the ?-MQRL given by

where

Applying straightforward algebra, it can be written as

in which F_{i}?¹(p;x_{(-i)})=inf{x_{i}:F(x)=p} and vector x_{(-i)} has dimension n-1 and is obtained by removing the i^{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 (2017)).

<lemma/>Consider the reliability function F with the hazard rate function ?(x), and the ?-MQRL function q_{?}(x), then for every i=1,2,…,n we have

and in turn

Dynamic multivariate ?-quantile residual life

Let F represent the reliability function of a bivariate random variable X. For brief representation, denote lim_{??0}(1/?)P(x?<X??x?+?, X?>x?) by P_{?1}(X?=x?, X?>x?) and similarly lim by. The conditional hazard rate functions of X are (cf. Shaked and Shanthikumar (1991b) or Cox (1972))

and Intuitively ?_{i}(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 (1972)). Shaked and Shanthikumar (1991b) applied the conditional hazard rate functions in description attributes of dynamic bivariate MRL. We define the dynamic ?-BQRL functions by

andwhereandSimple calculations implies

andand the expressions for F??¹(p;x) and F?^{?-1}(p;x?) are analogous.

Relations (<ref>96011802</ref>) to (<ref>96011804</ref>) 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 which 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 ?-BQRL.

<theorem/>Let q_{?,i}(x?,x?), i=1,2, have continuous differentiation functions with respect to their both coordinates. Then, we have

Proof. Due to the relation q_{?,1}(x)=q_{?,1}(x,x), the differentiation of q_{?,1}(x) can be described as the sum of differentiations in two directions

Taking n=2 and applying (<ref>96011801</ref>) with i=1, gives the first expression. By some algebra for the second differentiation, we have

which shows (<ref>96011809</ref>). Analogous statements indicate (<ref>96011806</ref>). To justify (<ref>95121</ref>), we consider the equation

By differentiation from y with respect to x in the equation inside brackets and applying definition of q_{?,1}(x | x?) the result follows. In a similar way, (<ref>95122</ref>) obtains, and this complete the proof.< > ?

Suppose that X consists of two independent components X? and X?. It is easily detectable that q_{?,1}(x | x?)=q_{?,1}(x) and q_{?,2}(x | x?)=q_{?,2}(x) for every x? and x? where q_{?,i}(x) shows the ?-QRL defined by (<ref>96020801</ref>) for X_{i}. In the sequel, we will see that a form of positive dependency between X? and X? imply q_{?,1}(x)?q_{?,1}(x | x?) and q_{?,2}(x)?q_{?,2}(x | x?). Therefore, for X? and X? positively dependent in such way, (<ref>96011809</ref>) and (<ref>96011806</ref>) respectively imply

andFix x? and suppose that ??(x | x?) is increasing in x then (<ref>95121</ref>) implies that q_{?,1}(x | x?) decreases in x. Similar argument holds for q_{?,2}(x | x?).

Example 1. The reliability function

proposed by Marshal and Olkin (1967) reveals a simple bivariate structure. Direct calculations shows

andwhere exhibit constant ?-BQRL functions.

Next, we generalize the concept to multivariate setting. Let the non-negative random vector X=(X?, X?,…,X_{n}) accommodates distribution F, and h_{x} captures history of events related to n components up to time x, i.e.,

in which I={i?,i?,…,i_{k}} 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 h_{x} as above, then for any component j?I?, the conditional hazard rate function can be written as

which describes the probability of instant failure of component j at time x, given history h_{x}. For empty set I, we have initial hazard functions. Denote ?-quantile of a random variable X with reliability function F by Q_{?}(X)=F?¹(?). Then for j?I?, we define the ?-MQRL function at time x by

that can be simplified to

For a simple representation denote

Observing history h_{x}={X?;x,…,X_{n};x} at x, i.e. I=?, we have the initial ?-MQRL functions that can be calculated as

;theorem/;Assume that q_{?,i}(x?,…,x_{n}), i=1,2,…,n, defined by (;ref;960120;/ref;), have continuous differentiation functions with respect to their coordinates. If I=?, we have a system of n equations

for i=1,…,n. Also, if we fix the history at x by h_{x}={X_{J}=x_{J},X_{J?};x1} where x_{J};x1, then for any survived component i?J? at x we can write

represents instantaneous risk of failure of component i by time x+q_{?,i}(x | h_{x}), given history h_{x} up to time x and censored history of components J? except i after x.

Other terms of it can be obtained by differentiation from both sides of the following equality with respect to x_{j} at point x, for j=1,…,n, j?iwhich proves (;ref;960215;/ref;). To show (;ref;96012301;/ref;), we can differentiate from both sides of

with respect to x, and hence the proof is completed. ?

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 (2007), 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?_{st}Y, if for every x;0, F(x)?G(x). In the multivariate framework, X?_{st}Y when E?(X)?E?(Y) for all non-decreasing functions ?:R???R? for which these expectations exist.

As a stronger ordering, X is said to be smaller than Y in the hazard rate order, X?_{hr}Y, if

for all x??x?. Provided that X and Y be equipped with the hazard rate functions ?_{X} and ?_{Y} respectively, the condition (;ref;9601281;/ref;) is equivalent with the inequality ?_{X}(x)??_{Y}(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 time x, the history h_{x} is referred to be more severe than h_{x}, denoted as h_{x}?h_{x}, if:

(i) every failed component in h_{x} also be failed in h_{x},

(ii) for common failures in both h_{x} and h_{x}, the failures in h_{x} are earlier than failures in h_{x}.

More formally,

where I?J=?, 01?x_{I}?y_{I}?x1 and 01?x_{J}?x1. In view of these notations, X is defined to be smaller than Y in the hazard rate order, X?_{hr}Y, if for every x?0

whenever h_{x}^{Y}?h_{x}^{X}, j?(I?J)? that is j shows a component survived in both histories and ?_{j}^{X} and ?_{j}^{Y} are the multivariate hazard rate functions defined in (;ref;96020201;/ref;) corresponding to X and Y, respectively. This order is not reflexive, that is X?_{hr}X is not necessarily the case and implies a kind of positive dependency, namely hazard increasing upon failure (cf. Shaked and Shanthikumar (1990, 1993) and Belzunce et. al. (2009)).

In many problems, we may deal with situations in which some of X_{i}’s be identically zero. Without loss of generality, X can be rearranged such that just X_{i}, i=1,2,…,k be identically zero and the rest of the vector be absolutely continuous. Then, we say X?_{hr}X if (<ref>96021301</ref>) be true for j>k.

As a weaker order, X is said to be smaller than Y in the ?-quantile residual life order, denoted as X?_{?-q}Y, if for every x?0

Franco-Pereira et. al. (2010) proved that X?_{hr}Y iff for any ??(0,1) we have X?_{?-q}Y. Here, we define X to be smaller than Y in the ?-quantile residual life if order, X?_{?-q}Y, if

whenever h_{x}^{Y}?h_{x}^{X} and for all components j alive in both histories. Like multivariate hazard rate order, it is not reflexive too. In fact X?_{?-q}X shows a positive dependency between components of X. Situations in which some of X_{i}’s be identically zero can be treated as explained for multivariate hazard rate order.

;theorem/;;label;96020305;/label; For two vectors X and Y, X?_{hr}Y if and only if for every ??(0,1) we have X?_{?-q}Y.

Proof. Firstly, we show that X?_{hr}Y if and only if for every x?0, t?0, h_{x}^{Y}?h_{x}^{X} and j alive in both of them

To achieve this, let h_{x}^{X}={X_{I}=x_{I}, X_{J}=x_{J},X_{(I?J)?};x1} and h_{x}^{Y}={Y_{I}=x_{I}, Y_{I?};x1}. We notice that X?_{hr}Y iffwhich is equivalent with the statement that

be decreasing in the jth survived component at x. Denote {Y_{I}=x_{I},Y_{I?};x1, Y_{j};y} by h_{x}^{Y}j,y and assume similar notation h_{x}^{X}j,x. Now it is easy to check that (;ref;96020301;/ref;) is the case ifffor every x?0, t?0, which is apparently equivalent with (;ref;96020302;/ref;). Thus, the result follows by definition of ?-MQRL function in (;ref;960126;/ref;) and (;ref;96020302;/ref;), and the proof is completed. ?

Remark 1. Let x and two histories h_{x}?h_{x} be fixed. It can be seen from Shaked and Shanthikumar (1991b) that X?_{hr}Y implies

in which (X-x1)? is the vector ((X?-x)?,…,(X_{n}-x)?) and (X_{i}-x)?=max{X_{i}-x,0}, i=1,…,n. As can be seen from the proof of Theorem ;ref;96020305;/ref;, we can write

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_{x}. 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 x?0

where h_{x}?h_{x} and i is an alive component at time x in both histories. Apparently, it is equivalent to say that

Shaked and Shanthikumar (1991b) 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 (1990, 1991b).

;theorem/;;label;96020104;/label; A sufficient and necessary condition for X to be ?-QRL-DF is that

Proof. First, we state (;ref;96013101;/ref;) in terms of ?-MQRL of X. Rename vectors of the left hand side and right hand side of (;ref;96013101;/ref;) 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, (;ref;96013101;/ref;) is equivalent with

whence h_{y}^{?}?h_{y}^{?}, k?I?-{j} shows a component alive at time y at both histories h_{y}^{?} and h_{y}^{?} and y?0. On the other hand, we have

andThus, we can write (;ref;96020901;/ref;) in the form

for every y?0, h_{y}^{?}?h_{y}^{?} and k?I?-{i} which is alive in both histories. Clearly, (;ref;96020101;/ref;) implies (;ref;96020102;/ref;) which by the fact that I, j and x are arbitrary proves the necessity part.

To show sufficiency, assume (;ref;96013101;/ref;) holds and h_{t}?h_{t} be two arbitrary histories up to time t?0. We need to show that

for 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 h_{t}, to be more sever than h_{t}. 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 x_{I}, 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 h_{t} about component j:

(i) it be alive at time t,

(ii) it fail at a time x such that x?x?t.

Based on these arrangements, h_{t} and h_{t} can be written in one of the following cases (a) or (b).

In either case, (;ref;96013101;/ref;) implies that q_{?,k}^{X}(t | h_{t})?q_{?,k}^{X}(t | h_{t}) for every alive component k at time t in history h_{t}, and this completes the proof. ?

Remark 2. A random vector X is said to hold positive dependency property SL if

while I?{1,…,n}, j?I?, x?0 and 01?x_{I}?x1 (cf. Shaked and Shanthikumar (1990) for more information). Now by Theorem ;ref;96020305;/ref;, it is immediate that

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 X is said to accommodate decreasing ?-QRL if q_{?}(x) be decreasing in x?0 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 ;ref;96020603;/ref;, in the case that their lengths are not necessarily equal. Consider two histories h_{x} and h_{x?} with different lengths x?x?. Then, h_{x?} is referred to be more severe than h_{x}, notationally h_{x}?h_{x?}, if h_{x}?h_{x} where h_{x} contains all information of h_{x?} over 0,x.

Define a shift operator ?_{x} on a random vector X by ?_{x}X=(X-x1)?, x?0. For a random vector X, we extend the arguments (;ref;96020501;/ref;) and (;ref;96020502;/ref;) respectively by

for every arbitrary history h_{x} and

whence h_{x?}?h_{x??}. These extensions are similar with conditions for multivariate decreasing mean residual life by Shaked and Shanthikumar (1991b). Another possible extension of (;ref;96020502;/ref;) is

where h_{x} and h_{x?} coincide on the interval 0,x. This means that all failed components in h_{x} are failed in h_{x?} too with equal failure times. The condition (;ref;96020506;/ref;) is similar with the multivariate increasing failure rate, proposed in Arjas (1984), and a multivariate decreasing mean residual life notion of Shaked and Shanthikumar (1991b).

Here we need some further notations to prove the following result. For a;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 h_{t} 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} be alive at a in h_{a,b} too and if a component fail in history h_{a,b} at time x, it also fail in h_{a+u,b+u} at some time x?+u such that x??x.

Let 0?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/;;label;96020507;/label; Three conditions (;ref;96020504;/ref;), (;ref;96020505;/ref;) and (;ref;96020506;/ref;) are equivalent.

Proof. It is readily detectable that

for every x,s?0, h_{0,s}¹?h_{x,x+s}¹ and h_{x}, for components j alive at x+s in history h_{x,x+s}¹. Now we must show that X satisfies (;ref;96020505;/ref;), that is

for every x???x??0, t?0, ?_{x?}??_{x??} and h_{x?,x?+t}²?h_{x??,x??+t}².

Let t, x??, x?, ?_{x?}??_{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 h_{x} to be ?_{x?}?h_{x?,x?+t}², ?_{x??-x?,x??}?h_{x??,x??+t}² and ?_{0,x??-x?}. By these considerations, it can be easily checked that (;ref;96021001;/ref;) implies (;ref;96021002;/ref;), which completes the proof immediately. ?

In the presence of Theorem ;ref;96020507;/ref;, the following definition is well-defined.

Definition 1. A random vector X is decreasing multivariate ?-quantile residual life (D ?-MQRL) if one of the conditions (;ref;96020504;/ref;), (;ref;96020505;/ref;) or (;ref;96020506;/ref;) be the case.

If X satisfies the condition (;ref;96020504;/ref;), then apparently we have X?_{?-q}X. This means that the D ?-MQRL implies the positive dependency property ?-QRL-DF.

Shaked and Shanthikumar (1991a) proposed X to be multivariate increasing failure rate (MIFR) when

for every x??0 and history h_{x?}. It is a proper extension of the following condition

When it holds, X is said to accommodate an increasing failure rate form.

;theorem/;;label;960205012;/label; The random vector X is MIFR iff it is D ?-MQRL for every ??(0,1).

Proof: The proof follows immediately by (;ref;96020504;/ref;), (;ref;96020511;/ref;) and Theorem ;ref;96020305;/ref;. ?

Example 2. Ross (1984) 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, ?_{k}(t | h_{t}), reduces to

Applying Theorem 1 from Shaked and Shanthikumar (1991a), this model follows MIFR property and in turn D ?-MQRL for every ??(0,1) if

for every J??I??{1,2,…,n} where show alive components at t, and k?J?.

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 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 ?-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 descend of initial ?-QRL functions. As results show similar spirit holds in the multivariate case. A new multivariate stochastic order, namely ?-MQRL order, has been defined. The results reveals 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 lifetimes property discussed in Shaked and Shanthikumar (1990). 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 solve the following functional equations in terms of F

where

(ii) Provide a non-parametric estimator of the ?-MQRL functions when X_{i}, i=1,2,…,n, be a sample of n independent and identically distributed random vectors.

Declarations

Acknowledgement

The 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).

Competing interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ contributions

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

ltivariate Analysis, 84, 173-189.