Characterizations of the Weak Bivariate Failure Rate Order and Bivariate IFR Aging Class

Kayid, M.

doi:https://doi.org/10.1155/2022/2573667

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Abstract Introduction and Preliminaries Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 2573667 | https://doi.org/10.1155/2022/2573667

Characterizations of the Weak Bivariate Failure Rate Order and Bivariate IFR Aging Class

M. Kayid¹

Academic Editor: Antonio Di Crescenzo

Received29 Dec 2021

Revised15 Mar 2022

Accepted05 Apr 2022

Published30 Apr 2022

Abstract

In this paper, two characterizations of the weak bivariate failure rate order over the bivariate Laplace transform order of two-dimensional residual lifetimes are given. The results are applied to characterize the weak bivariate failure rate ordering of random pairs by the weak bivariate mean residual lifetime ordering of the minima of pairs with exponentially distributed random pairs with unspecified mean. Moreover, a well-known bivariate aging term, namely, the bivariate increasing failure rate, is characterized by the weaker bivariate decreasing mean residual lifetime property of a random pair of minima.

1. Introduction and Preliminaries

Consider the random pair as lifetimes of two devices with joint survival function (s.f.) . Let us assume that the first device is at age and the second device is at age . The first device and the second device are assumed to be working at the times and , respectively. The residual lifetime random pair is defined asin which with so that is the upper bound of the support of , and moreover

The couple is called the pair of current ages. The pair is called the bivariate residual lifetime associated with . The residual lifetime random pair is adopted to measure the time-to-failure of the devices or the system composed of these devices in light of the ages of the devices. The random variables , are the marginal random variables of . It can be seen that has joint s.f.where . If is assumed to be linked with an absolutely continuous joint cumulative distribution function (c.d.f.) with the corresponding joint probability density function (p.d.f.) , then the joint p.d.f. of is obtained by

If we assume that has marginal s.f. , then the p.d.f. of (when it exists) is denoted by . The conditional s.f. of and is obtained, respectively, asand the corresponding conditional p.d.f.s are, therefore, revealed as

In the case when and are independent, the random variable defined in (2) is equal in distribution with , which is the commonly used univariate residual lifetime which is known in the literature.

The failure gradient of is given byin which

Denote also by the bivariate mean residual lifetime (m.r.l.) function of asin which

In the following, two well-known bivariate stochastic orders are defined, which are frequently used in the sequel.

Definition 1. Let and be two non-negative random pairs s.f.s and and failure gradients and , respectively. Then, it is said that is smaller than in weak bivariate failure rate order (denoted as ) whenever , for all , or equivalently ifFor further properties of bivaraite/multivariate failure rate functions and also some of their applications in different contexts, we refer the readers to Basu [1]; Johnson and Kotz [2]; Marshal [3]; Navarro and Ruiz [4]; Khaledi and Kochar [5]; Misra and Naqvi [6]; Badía et al. [7]; Badía and Lee [8]; Nair and Vinesh [9]; and Gupta and Kirmani [10].

Definition 2. Let and be two non-negative random pairs with s.f.s and and m.r.l. functions and , respectively. We then say that is smaller than in weak bivariate mean residual lifetime order (denoted as ) if , for all , or equivalently ifNote that implies that (cf. Hu et al. [11]).
For any non-negative random variable with c.d.f. , the Laplace–Stieltjes transform is defined byEvidently, is non-increasing in . Let denote the s.f. of which measures the probabilities of selective right tails of the distributor of . The Laplace transform of is given byIt is worth mentioning that the Laplace transforms involved in (13) and (14) always exist (since is non-negative). The Laplace transform has been used to be regarded as actuarial amounts, such as indemnities associated with risks, incomes associated with financial transactions, or life premiums in life insurance (see Belzunce et al. [12]; Denuit [13]; and Belzunce et al. [14]). It can be easily verified that if is absolutely continuous, thenIn view of (13) and also in the spirit of (14), one can rewrite and where stands for a random variable with exponential distribution with mean .
Given two random variables and with Laplace transforms and , respectively, it is said that is smaller than in Laplace transform order (denoted by ) whenever , for all . Equivalently, from (15),Let us assume that , and , denote the residual lifetime of and , respectively, where and denote the upper bound of supports of and . Belzunce et al. [12] applied the Laplace transform to compare residual lifetimes in place of the original random variables. We say that with p.d.f. and s.f. is smaller than with p.d.f. and s.f. in the failure rate order (denoted as ) if , for all , in which , and , are the failure rate functions of and , respectively. Belzunce et al. [15] showed thatIt is said that with m.r.l. function given by is smaller than with m.r.l.f. given by (denoted as ) whenever , for all . Belzunce et al. [15] also proved thatFor the non-negative random pair with joint p.d.f. , the bivariate Laplace–Stieltjes transform is given bySuppose that and have respective Laplace transforms and . Then, it is said that is smaller than in the bivariate Laplace transform (denoted by ) whenever , for all . From Theorem 7.D.1. of Shaked and Shanthikumar [16], it is deduced that implies , but the reversed implication is not true in general.
The aim of this paper is to compare the characterizations from (17) and (18) given characterizations to bivariate distributions, enumerating also the dependence structure of random pairs as a new aspect in ordering bivariate distributions. By developing the theory for the bivariate lifetime distribution, it is found that the dependence structure of the random pairs considered has no influence on the characterization of the weak bivariate failure rate order and also on a characterization of a bivariate increasing failure rate property, as will be shown.
The paper is organized as follows. Section 2 presents the main results of the paper, which include a characterization of the weak bivariate failure rate order using the univariate Laplace transform order applied to conditional bivariate residual lifetimes, and then a development of this characterization to a stronger case. In Section 3, in the spirit of the previous characterizations, we present further characterizations of the weak bivariate failure rate order in terms of the weak bivariate mean residual life order and also a characterization result for the increasing bivariate failure rate class using the decreasing bivariate mean residual life. In Section 4, we conclude the paper with a detailed summary of the paper and also a future perspective of the developments of the results on the multivariate cases.

2. Characterizations of Weak Bivariate Failure Rate Order

In this section, the bivariate Laplace transform is applied to residual lifetime pair to compare the residual lifetimes of and . Denote by , the Laplace transform of the conditional random variable and denote by the Laplace transform of given which is randomly drawn. By some routine calculation, one obtainswhere

To derive the Laplace transform of bivariate residual lifetime first, let us get the expression of the Laplace transform of . The formulas given in (20) and (21) can be fulfilled by the identities (5) to get

Now, one can develop that

The following result presents equivalent conditions for the Laplace transform ordering of and .

Proposition 1. Let and be two non-negative random pairs having joint s.f.s and with bivariate residual lifetimes and , respectively. Then,(i), for all if, and only if, is non-decreasing in for every .(ii), for all if, and only if, is non-decreasing in for every .

Proof. We prove only assertion (i). Assertion (ii) can be proved by an analogous method. It is easily verified that for all and ,in which stands for equality in sign. Thus, it follows that for all ,if and only if (see (22))This is also equivalent to .
Suppose that with where denotes an exponentially distributed random variable with mean . Let us assume that and are independent and, further, and are also independent. Denote by the minimum of and . The random pair has joint s.f.Likewise, provided that and are also independent, has joint s.f. . From definition of bivariate m.r.l. and by Proposition 1,Therefore, applying the order,In the next result, the order is characterized by the Laplace transform order of residual lifetimes of marginal distributions of and .

Theorem 1. Let and be two non-negative random pairs having joint s.f.s and , respectively. Then,

Proof. We first prove that implies that . It is known that, for every ,Using the identities given in (31), the identities given in (22), and by applying (6), one can getand similarly,From Proposition 1, it is deduced that , for all , holds if, and only if, (32) is non-decreasing in for every and for all , and in particular, when , , for all , concludes that (32) is non-decreasing in for every . In parallel, , for all , holds if, and only if, (33) is non-decreasing in for every and for all , and in the special case when , , for all , implies that (33) is non-decreasing in for every . Notice that for all and, also, for all ,in which is an integrable function for all , sinceBy Lebesgue’s dominated convergence theorem, we havewhere . In a similar manner,where . Thus,Further,Making use of (38) together with (39) concludes that . To prove the reversed implication, we assume that . Select an arbitrary and fix it and also fix as an arbitrary value. Define the functions and asrespectively, so that and . From assumption, is non-decreasing in . Hence, is TP₂ in , and also it is plain to see that is also TP₂ in . By general composition theorem of Karlin [17], is also TP₂ in . Since the conclusion is not affected by the choice of and also , one concludes that is non-decreasing in , for all and for every . By Proposition 1 (i), it follows that , for all . By a similar discussion, it is realized that also implies that . The proof is completed.
The following example illustrates an application of Theorem 1.

Example 1. Let follow the following s.f. associated with a mixture distribution:and, furthermore, let follow a mixture distribution having s.f.where and is an arbitrary value in and also for every . By appealing to routine calculations, the bivariate failure rates of and are specified, respectively, byfor every . We also can getfor every . We can observe that if and so that , then one realizes that for all and for every , i.e., , and also it is seen that , for every and for all and for all , which means that , for all . Thus, the result of Theorem 1 is confirmed by making ordering conditions on parameters of a parametric distribution.
By Theorem 7.D.5 in Shaked and Shanthikumar [16], implies . From Theorem 1, we can, therefore, deduce that if , for all , then . One may question whether the converse implication holds true. We show that the reversed implication is also satisfied.

Proposition 2. implies , for all .

Proof. By definition, the proof is obtained if we show that , for all and for all . From assumption, holds. By Theorem 1, it follows that , and thus , for all and for every . We prove using this ordering condition thatBy applying (21), the expected result will be secured. We can getwhere and is the Laplace transform of which can be obtained by substituting (5) when applied on into (22). Now, we develop thatwhere for , , and . In a similar fashion,The assumption implies that for all and for every as mentioned earlier. Thus,In addition,Now, we can conclude thatThus, Theorem 1 together with Proposition 2 provide the following result as a characterization property of the order by bivariate Laplace transform ordering of residual lives. The proof is explicit, and thus we omit it.

Theorem 2. Let and be two non-negative random pairs having joint s.f.s and , respectively. Then,

3. Characterizations of Bivariate Increasing Failure Rate Aging Class

One of the attractive applications of stochastic orderings in the univariate, bivariate, and multivariate settings is the characterization of an aging class of lifetime distributions. This advantage arises from the stochastic comparison of the remaining lifetimes of units after a succession of ages.

In this section, we provide further characterizations of the order between lifetime pairs of and by means of the order of typical transformations of and . Descriptions of some bivariate aging notions in terms of the Laplace transform ordering of residual lives of and a characterization result for a bivariate IFR aging property using a weaker bivariate DMRL aging behavior are given.

Theorem 3. Let and be two lifetime random pairs with s.f.s and , respectively. Suppose that is a non-negative random pair which is independent of both and . Then,

Proof. Denote by the joint s.f of . Suppose that and have respective joint s.f.s and which are given, from assumption, aswhere . Trivially, is equivalent to . Thus, . Conversely, since for all non-negative random pairs which are independent from and , we have , by considering where and are independent exponential random variables, one has , for all . From definition, this leads to the inequalities (55) and (56) given below being satisfied for all asandIt is evident that (55) is equivalent to and also (56) is equivalent to both of which hold for all . Therefore, in the spirit of Theorem 1, it follows that . This completes the proof.
The result of Theorem 1 can be used to characterize the order between and using the order when applying on and as two dependent random pairs on .

Theorem 4. Let and be two lifetime random pairs with s.f.s and , respectively. Then,

Proof. We first prove that implies for every non-negative . If and if we denote , and , then since and are increasing in and , respectively, by Theorem 6.D.4. of Shaked and Shanthikumar [16], , and therefore . To prove the converse part, note that and have respective s.f.sin which and . Therefore,and moreover,By taking where , (59) implies that for all ,and, in parallel, (60) concludes for all thatHence, applying Theorem 1, the result follows.
The following classes of bivariate aging notions are quite well known in the literature (see Bassan et al. [18] and Lai and Xie [19]).

Definition 3. Suppose that is a random pair of lifetimes which has joint s.f. , joint failure rate function , and joint m.r.l. function . Then,(i) is said to have bivariate increasing failure rate (BIFR) aging property whenever is non-increasing in for every or equivalently if , for all and , .(ii) is said to have bivariate decreasing mean residual lifetime (BDMRL) aging property provided that , for all and , .Rajesh et al. [20] introduced and studied some notions of aging based on the Laplace transform of bivariate residual life. It is well known that the BIFR property implies the BDMRL property, but the converse is not true in general. In the next result, we characterize the BIFR class of joint lifetime distributions by means of the BDMRL property.

Theorem 5. Suppose that is a non-negative random pair. Then,

Proof. It can be readily seen that is BIFR if, and only if, for every . This together with Theorem 3 concludes that is BIFR if, and only if, for all . It is not hard to prove thatTherefore, is BIFR if, and only if, for all which, equivalently, holds whenever is BDMRL for every

4. Conclusions

Two objectives were pursued with this work. It is well known in the literature that the bivariate Laplace transform order and the weak bivariate mean residual life order are both weaker than the bivariate weak failure rate order in the sense that the latter implies the former ones. The first objective of this work was to characterize the bivariate weak failure rate order of two pairs of random lifetimes using the bivariate Laplace transform order applied to the residual lifetimes. The result provides further characterizations of the bivariate weak failure rate order between two random pairs of lifetimes by the weak bivariate mean residual lifetime order applied to certain classes of lifetime transformations. One of these is the minimum between lifetimes and an independent random lifetime, and another is the class of distribution functions of the exponential distribution with an unspecified mean. The second goal of the paper was to characterize a known aging property, namely, the bivariate increasing failure rate, using a weaker aging property, namely, the bivariate decreasing mean residual life property, of a specified transformation of the underlying random pair of lifetimes. This class is the minimum of the underlying random lifetimes with a random pair of independent exponential variables, each having an arbitrary mean.

In the future, the author would like to see whether the given characterizations extend to higher (more than two) dimensions. Consider two random vectors and of lifetimes with partially differentiable s.f.s and , respectively. According to Hu et al. [11], it is said that is smaller than in the weak multivariate failure rate order (denoted by ) whenever is non-decreasing in . It is also said that is smaller than in the multivariate Laplace transform order (denoted by ) if , for all (see, e.g., Shaked and Shanthikumar [16]). Let us consider and as the multivariate residual lifetime vectors associated with and , respectively, where is a vector of time points in . We can also consider the marginal conditional residual lifetimes and for every . The result of Theorem 1 may be extended to the case where , in the multivariate case, is equivalent to , for all where . To develop the result of Theorem 2, we can also study whether , in the multivariate setting, is equivalent to , for all where . What success can be achieved depends on whether the techniques used in the bivariate setting can also be used in the multivariate case. The author is confident that the developments described above are feasible and can be presented in a future research work.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

This work was supported by Researchers Supporting Project number (RSP-2022/392), King Saud University, Riyadh, Saudi Arabia.

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Copyright © 2022 M. Kayid. 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.

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