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

Authors: M. Kayid
Journal: Journal of Mathematics
 (2022)

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## 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.


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## Body

## 1. Introduction and Preliminaries

Consider the random pairX⊤=X1,X2 as lifetimes of two devices with joint survival function (s.f.) F¯. Let us assume that the first device is at age t1 and the second device is at age t2. The first device and the second device are assumed to be working at the times t1 and t2, respectively. The residual lifetime random pair is defined as(1)X⊤t≔X1t,X2t=X1−t1,X2−t2|X1>t1,X2>t2,in which t=t1,t2 with ti<uXi so that uXi=supx≥0|FXix<1 is the upper bound of the support of Xi,i=1,2, and moreover(2)Xit=Xi−ti|X1>t1,X2>t2,∀t1,t2:F¯t1,t2>0.The couplet⊤=t1,t2 is called the pair of current ages. The pair Xt is called the bivariate residual lifetime associated with X=X1,X2. 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 Xit,i=1,2, are the marginal random variables of Xt. It can be seen that Xt has joint s.f.(3)F¯tx=F¯x1+t1,x2+t2F¯t1,t2,where x=x1,x2. If F¯ 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.) f, then the joint p.d.f. of Xt is obtained by(4)ftx=fx1+t1,x2+t2F¯t1,t2.If we assume thatXi has marginal s.f. F¯Xi, then the p.d.f. of Xi (when it exists) is denoted by fXi,i=1,2. The conditional s.f. of X1t and X2t is obtained, respectively, as(5)F¯X1tx1=F¯x1+t1,t2F¯t1,t2,F¯X2tx1=F¯t1,x2+t2F¯t1,t2,and the corresponding conditional p.d.f.s are, therefore, revealed as(6)fX1tx1=−∂/∂x1F¯x1+t1,t2F¯t1,t2,fX2tx2=−∂/∂x2F¯t1,x2+t2F¯t1,t2.In the case whenX1 and X2 are independent, the random variable Xit defined in (2) is equal in distribution with Xi−ti|Xi>ti,i=1,2, which is the commonly used univariate residual lifetime which is known in the literature.The failure gradient ofX is given by(7)hXt=−ΔlnF¯t=h1,Xt,h2,Xt,Δ=∂∂t1,∂∂t2,in which(8)h1,Xt=limδ⟶0+PX1t∈0,δδ=−∂∂t1lnF¯t,h2,Xt=limδ⟶0+PX2t∈0,δδ=−∂∂t2lnF¯t.Denote also bymX the bivariate mean residual lifetime (m.r.l.) function of X as(9)mXt=m1,Xt,m2,Xt,in which(10)m1,Xt=EX1t=∫t1+∞F¯x1,x2dx1F¯t1,t2,m2,Xt=EX2t=∫t2+∞F¯x1,x2dx2F¯F¯t1,t2.In the following, two well-known bivariate stochastic orders are defined, which are frequently used in the sequel.Definition 1.
LetX and Y be two non-negative random pairs s.f.s F¯ and G¯ and failure gradients hX and hY, respectively. Then, it is said that X is smaller than Y in weak bivariate failure rate order (denoted as X≤Ywfr) whenever hi,Xt≥hi,Yt,i=1,2, for all t∈ℝ+2, or equivalently if(11)G¯tF¯tis non−decreasing int=t1,t2∈t∈ℝ+2:G¯t>0.
For 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.
LetX and Y be two non-negative random pairs with s.f.s F¯ and G¯ and m.r.l. functions mX and mY, respectively. We then say that X is smaller than Y in weak bivariate mean residual lifetime order (denoted as X≤Ywmrl) if mi,Xt≤mi,Yt,i=1,2, for all t∈ℝ+2, or equivalently if(12)∫t1+∞G¯x1,x2dx1∫t1+∞F¯x1,x2dx1is non−decreasing int1≥0,∫t2+∞G¯x1,x2dx2∫t2+∞F¯x1,x2dx2is non−decreasing int2≥0.
Note thatX≤Ywfrimplies that X≤Ywmrl (cf. Hu et al. [11]).
For any non-negative random variableX with c.d.f. FX, the Laplace–Stieltjes transform is defined by(13)LXs≔∫0+∞e−sxdFXx,s>0.
Evidently,LXs is non-increasing in s. Let F¯X≡1−FX denote the s.f. of X which measures the probabilities of selective right tails of the distributor of X. The Laplace transform of F¯X is given by(14)LX∗s≔∫0+∞e−sxF¯Xxdx,s>0.
It is worth mentioning that the Laplace transforms involved in (13) and (14) always exist (since X 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 FX is absolutely continuous, then(15)LX∗s=1s1−LXs,s>0.
In view of (13) and also in the spirit of (14), one can rewrite LXs=Ee−sX and LX∗s=EminX,Es where Es stands for a random variable with exponential distribution with mean 1/s.
Given two random variablesX and Y with Laplace transforms LX and LY, respectively, it is said that X is smaller than Y in Laplace transform order (denoted by X≤YLt) whenever LXs≥LYs, for all s≥0. Equivalently, from (15),(16)X≤YLt⟺LX∗s≤LY∗s,for alls>0.
Let us assume thatXt=X−t|X>t,t<uX, and Yt=Y−t|Y>t,t<uY, denote the residual lifetime of X and Y, respectively, where uX and uY denote the upper bound of supports of X and Y. Belzunce et al. [12] applied the Laplace transform to compare residual lifetimes in place of the original random variables. We say that X with p.d.f. fX and s.f. F¯X is smaller than Y with p.d.f. fY and s.f. F¯Y in the failure rate order (denoted as X≤Yfr) if hXt≥hYt, for all t<minuX,uY, in which hXt=fXt/F¯Xt,t<uX, and hYt=fYt/F¯Yt,t<uY, are the failure rate functions of X and Y, respectively. Belzunce et al. [15] showed that(17)X≤Yfr⟺Xt≤YLtt,for allt<minuX,uY.
It is said thatX with m.r.l. function mX given by mXt=EXt is smaller than Y with m.r.l.f. mY given by mYt=EYt (denoted as X≤Ymrl) whenever mXt≤mYt, for all t<minuX,uY. Belzunce et al. [15] also proved that(18)X≤Yfr⟺1−e−sX≤1mrl−e−sY,for alls>0.
For the non-negative random pairX=X1,X2 with joint p.d.f. fx1,x2, the bivariate Laplace–Stieltjes transform is given by(19)LXs=∫0+∞∫0+∞e−s1X1−s2X2fx1,x2dx1dx2,s=s1,s2∈ℝ+2.
Suppose thatX⊤=X1,X2 and Y⊤=Y1,Y2 have respective Laplace transforms LX and LY. Then, it is said that X is smaller than Y in the bivariate Laplace transform (denoted by X≤YBLt) whenever LXs≥LYs, for all s=s1,s2∈ℝ+2. From Theorem 7.D.1. of Shaked and Shanthikumar [16], it is deduced that X≤YBLt implies Xi≤YLti,i=1,2, 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. Section2 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 pairXt to compare the residual lifetimes of X and Y. Denote by LX2|X1=x1s2,s2>0, the Laplace transform of the conditional random variable X2|X1=x1 and denote by LX2|X1s2 the Laplace transform of X2 given X1 which is randomly drawn. By some routine calculation, one obtains(20)LXs≔Ee−s1X1Ee−s2X2|X1=Ee−s1X1LX2|X1s2=1−s1LX1∗s1−s2LX2∗s2+s1s2LX∗s,si>0,i=1,2,where(21)LX∗s=∫0+∞∫0+∞e−s1x1−s2x2F¯x1,x2dx1dx2.To derive the Laplace transform of bivariate residual lifetimeXt=X1t,X2t first, let us get the expression of the Laplace transform of Xit,i=1,2. The formulas given in (20) and (21) can be fulfilled by the identities (5) to get(22)LX1t∗s1≔∫t1+∞e−s1x1F¯x1,t2dx1e−s1t1F¯t1,t2,LX2t∗s2≔∫t2+∞e−s2x2F¯t1,x2dx2∫t2+∞e−s2x2F¯t1,x2dx2e−s2t2F¯t1,t2.Now, one can develop that(23)LXts≔1−s1∫t1+∞e−s1x1F¯x1,t2dx1e−s1t1F¯t1,t2−s2∫t1+∞e−s2x2F¯t1,x2dx2e−s2t2F¯t1,t2+s1s2∫t1+∞∫t2+∞e−s1x1−s2x2F¯x1,x2dx2dx1F¯x1,t2.The following result presents equivalent conditions for the Laplace transform ordering ofXit and Yit,i=1,2.Proposition 1.
LetX⊤=X1,X2 and Y⊤=Y1,Y2 be two non-negative random pairs having joint s.f.s F¯ and G¯ with bivariate residual lifetimes X⊤t=X1t,X2t and Y⊤t=Y1t,Y2t, respectively. Then,(i)
X1t≤YLt1t, for all t∈ℝ+2 if, and only if, ∫t1+∞e−s1x1G¯x1,t2dx1/∫t1+∞e−s1x1F¯x1,t2dx1 is non-decreasing in t1≥0 for every t2≥0.(ii)
X2t≤YLt2t, for all t∈ℝ+2 if, and only if, ∫t2+∞e−s2x2G¯t1,x2dx2/∫t2+∞e−s2x2F¯t1,x2dx2 is non-decreasing in t2≥0 for every t1≥0.Proof.
We prove only assertion (i). Assertion (ii) can be proved by an analogous method. It is easily verified that for allti≥0,i=1,2 and s1≥0,(24)∂∂t1∫t1+∞e−s1x1G¯x1,t2dx1∫t1+∞e−s1x1F¯x1,t2dx1=sgn+e−s1t1F¯t1,t2∫t1+∞e−s1x1G¯x1,t2dx1−e−s1t1G¯t1,t2∫t1+∞e−s1x1F¯x1,t2dx1,in which =sign stands for equality in sign. Thus, it follows that for all ti≥0,i=1,2,(25)∂∂t1∫t1+∞e−s1x1G¯x1,t2dx1∫t1+∞e−s1x1F¯x1,t2dx1≥0,∀s1≥0,if and only if (see (22))(26)LX1t∗s1=∫t1+∞e−s1x1F¯x1,t2dx1e−s1t1F¯t1,t2≤∫t1+∞e−s1x1G¯x1,t2dx1e−s1t1G¯t1,t2=LY1t∗s1,∀s1≥0.
This is also equivalent toX1t≤YLt1t.
Suppose thatTs=Ts1,Ts2 with s=s1,s2 where Tsi denotes an exponentially distributed random variable with mean 1/si,i=1,2. Let us assume that X and Ts are independent and, further, Ts1 and Ts2 are also independent. Denote by a∧b the minimum of a and b. The random pair X∧Ts=X1∧Ts1,X2∧Ts2 has joint s.f.(27)F¯s∗x=PX∧Ts>x=PX>x,Ts>x=e−s1x1−s2x2F¯x,x=x1,x2∈ℝ2.
Likewise, provided thatY and Ts are also independent, Y∧Ts=Y1∧Ts1,Y2∧Ts2 has joint s.f. G¯sx=e−s1x1−s2x2G¯x. From definition of bivariate m.r.l. and by Proposition 1,(28)X1t≤YLt1t⟺m1,X∧Tst1,t2≤m1,Y∧Tst1,t2,X2t≤YLt2t⟺m2,X∧Tst1,t2≤m2,Y∧Tst1,t2.
Therefore, applying thewmrl order,(29)Xit≤YLtit,i=1,2⟺X∧Ts≤Ywmrl∧Ts,for alls∈ℝ+2.
In the next result, thewfr order is characterized by the Laplace transform order of residual lifetimes of marginal distributions of Xt and Yt.Theorem 1.
LetX⊤=X1,X2 and Y⊤=Y1,Y2 be two non-negative random pairs having joint s.f.s F¯ and G¯, respectively. Then,(30)X≤Ywfrif, and only if,Xit≤YLtit,i=1,2,for allt∈ℝ+2.Proof.
We first prove thatXit≤YLtit,i=1,2,for allt∈ℝ+2 implies that X≤Ywfr. It is known that, for every i=1,2,(31)LXit∗si=1si1−LXitsi,LYit∗si=1si1−LYitsi.
Using the identities given in (31), the identities given in (22), and by applying (6), one can get(32)∫t1+∞e−s1x1G¯x1,t2dx1∫t1+∞e−s1x1F¯x1,t2dx1=G¯t1,t2LY1t∗s1F¯t1,t2LX1t∗s1=G¯t1,t2−G¯t1,t2∫0+∞e−s1x1fX1tx1dx1F¯t1,t2−F¯t1,t2∫0+∞e−s1x1fY1tx1dx1=G¯t1,t2−∫t1+∞e−s1x1−t1−∂/∂x1G¯x1,t2dx1F¯t1,t2−∫t1+∞e−s1x1−t1−∂/∂x1F¯x1,t2dx1,and similarly,(33)∫t2+∞e−s2x2G¯t1,x2dx2∫t2+∞e−s2x2F¯t1,xt2dx2=G¯t1,t2LY2t∗s2F¯t1,t2LX2t∗s2=G¯t1,t2−G¯t1,t2∫0+∞e−s2x2fX2tx2dx2F¯t1,t2−F¯t1,t2∫0+∞e−s2x2fY2tx2dx2=G¯t1,t2−∫t2+∞e−s2x2−t2−∂/∂x2G¯t1,x2dx2F¯t1,t2−∫t2+∞e−s2x2−t2−∂/∂x2F¯t1,x2dx2.
From Proposition1, it is deduced that X1t≤YLt1t, for all t∈ℝ+2, holds if, and only if, (32) is non-decreasing in t1 for every t2≥0 and for all s1≥0, and in particular, when s1⟶+∞, X1t≤YLt1t, for all t∈ℝ+2, concludes that (32) is non-decreasing in t1 for every t2≥0. In parallel, X2t≤YLt2t, for all t∈ℝ+2, holds if, and only if, (33) is non-decreasing in t2 for every t1≥0 and for all s2≥0, and in the special case when s2⟶+∞, X2t≤YLt2t, for all t∈ℝ+2, implies that (33) is non-decreasing in t2 for every t1≥0. Notice that for all s1≥0 and, also, for all t∈ℝ+2,(34)e−s1x1−t1Ix1>t1−∂∂x1F¯x1,t2<−∂∂x1F¯x1,t2,in which −∂/∂x1F¯x1,t2 is an integrable function for all t∈ℝ+2, since(35)∫0+∞−∂∂x1F¯x1,t2dx1=F¯X2t2≤1.
By Lebesgue’s dominated convergence theorem, we have(36)lims1⟶+∞∫t1+∞fx1,t2e−s1x1−t1dx1=lims1⟶+∞∫0+∞fx1,t2e−s1x1−t1Ix1>t1dx1,=∫0+∞lims1⟶+∞fx1,t2e−s1x1−t1Ix1>t1dx1=0,where fx1,t2=−∂/∂x1F¯x1,t2. In a similar manner,(37)lims1⟶+∞∫t1+∞gx1,t2e−s1x1−t1dx1=0,where gx1,t2=−∂/∂x1G¯x1,t2. Thus,(38)X1t≤YLt1t,∀t∈ℝ+2yieldsG¯t1,t2F¯t1,t2is non−decreasing int1,∀t2≥0.
Further,(39)X2t≤YLt2t,∀t∈ℝ+2yieldsG¯t1,t2F¯t1,t2is non−decreasing int2,∀t1≥0.
Making use of (38) together with (39) concludes that X≤Ywfr. To prove the reversed implication, we assume that X≤Ywfr. Select an arbitrary t2≥0 and fix it and also fix s1≥0 as an arbitrary value. Define the functions J and H as(40)t1,x1↦Jt1,x1=e−s1x1Ix1>t1,t1,x1≥0,x1,i↦Hx1,i,i=1,2,x1≥0,respectively, so that Hx1,1=F¯x1,t2 and Hx1,2=G¯x1,t2. From assumption, G¯x1,t2/F¯x1,t2 is non-decreasing in x1≥0. Hence, H is TP2 in x1,i∈ℝ+×1,2, and also it is plain to see that J is also TP2 in t1,x1∈ℝ+×ℝ+. By general composition theorem of Karlin [17], ∫0+∞Jt1,x1Hx1,idx1 is also TP2 in t1,i∈ℝ+×1,2. Since the conclusion is not affected by the choice of t2≥0 and also s1≥0, one concludes that ∫t1+∞e−s1x1G¯x1,t2dx1/∫t1+∞e−s1x1F¯x1,t2dx1 is non-decreasing in t1≥0, for all s1≥0 and for every t2≥0. By Proposition 1 (i), it follows that X1t≤YLt1t, for all t∈ℝ+2. By a similar discussion, it is realized that X≤Ywfr also implies that X2t≤YLt2t. The proof is completed.
The following example illustrates an application of Theorem1.Example 1.
LetX⊤=X1,X2 follow the following s.f. associated with a mixture distribution:(41)F¯t=pe−λ1t1+t2+qe−λ1′t1+t2,t=t1,t2∈ℝ+2,and, furthermore, let Y=Y1,Y2 follow a mixture distribution having s.f.(42)G¯t=pe−λ2t1+t2+qe−λ2′t1+t2,t=t1,t2∈ℝ+2,where q=1−p and p is an arbitrary value in 0,1 and also λi,λi′>0 for every i=1,2. By appealing to routine calculations, the bivariate failure rates of X and Y are specified, respectively, by(43)hi,Xt=pλ1+qλ1′eλ1−λ1′t1+t2p+qeλ1−λ1′t1+t2,hi,Yt=pλ2+qλ2′eλ2−λ2′t1+t2p+qeλ2−λ2′t1+t2,for every i=1,2. We also can get(44)LXit∗si=p/si+λ1+q/si+λ1′eλ1−λ1′t1+t2p+qeλ1−λ1′t1+t2,LYit∗si=p/si+λ2+q/si+λ2′eλ2−λ2′t1+t2p+qeλ2−λ2′t1+t2,for every i=1,2. We can observe that if λ1≥λ2 and λ1′≥λ2′ so that λ1−λ1′λ2−λ2′≥0, then one realizes that hi,Xt≥hi,Yt for all t∈ℝ+2 and for every i=1,2, i.e., X≤Ywfr, and also it is seen that LXit∗si≤LYit∗si, for every i=1,2 and for all t∈ℝ+2 and for all si>0,i=1,2, which means that Xit≤YLtit,i=1,2, for all t∈ℝ+2. 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], Xt≤YBLtt implies Xit≤YLtit,i=1,2. From Theorem 1, we can, therefore, deduce that if Xt≤YBLtt, for all t∈ℝ+2, then X≤Ywfr. One may question whether the converse implication holds true. We show that the reversed implication is also satisfied.Proposition 2.
X≤Ywfr implies Xt≤YBLtt, for all t∈ℝ+2.Proof.
By definition, the proof is obtained if we show thatLXts≥LYts, for all s∈ℝ+2 and for all t∈ℝ+2. From assumption, X≤Ywfr holds. By Theorem 1, it follows that X2t≤YLt2t, and thus 1−s2LX2t∗s2≥1−s2LY2t∗s2, for all t∈ℝ+2 and for every s2≥0. We prove using this ordering condition that(45)s2LXt∗s−LX1t∗s1≥s2LYt∗s−LY1t∗s1,∀t∈ℝ+2,∀si≥0,i=1,2.
By applying (21), the expected result will be secured. We can get(46)LXt∗s=∫t1+∞∫t2+∞e−s1x1−s2x2F¯x1,x2dx2dx1e−s1t1−s2t2F¯t1,t2=es1t1∫t1+∞e−s1x1∫t2+∞e−s2x2F¯x1,x2dx2e−s2t2F¯x1,t2F¯x1,t2F¯t1,t2dx1=es1t1∫t1+∞e−s1x1LX2t∗∗s2F¯x1,t2F¯t1,t2dx1,where t∗=x1,t2 and LX2t∗∗s2 is the Laplace transform of F¯X2t∗ which can be obtained by substituting (5) when applied on X2t∗=X2−t2|X1>x1,X2>t2 into (22). Now, we develop that(47)DXt,s=LX1t∗s1−s2LXt∗s=∫t1+∞e−s1x1F¯x1,t2dx1e−s1t1F¯t1,t2−s2∫t1+∞∫t2+∞e−s1x1−s2x2F¯x1,x2dx2dx1F¯x1,t2=es1t1∫t1+∞e−s1x11−s2LX2t∗∗s2F¯x1,t2F¯t1,t2dx1=es1t1∫t1+∞e−s1x1LX2t∗s2F¯x1,t2F¯t1,t2dx1,where LX2t∗s2=Ee−s2X2t∗ for s2≥0, t2≥0, and x1≥0. In a similar fashion,(48)DYt,s=LY1t∗s1−s2LYt∗s=es1t1∫t1+∞e−s1x1LY2t∗s2G¯x1,t2G¯t1,t2dx1.
The assumptionX≤Ywfr implies that X2t∗≤YLt2t∗ for all x1≥0 and for every t2≥0 as mentioned earlier. Thus,(49)X≤YwfrgivesLY2t∗s2≥LX2t∗s2,∀x1≥0,∀t2,∀s2≥0.
In addition,(50)X≤YwfryieldsG¯x1,t2G¯t1,t2−F¯x1,t2F¯t1,t2≥0,∀x1>t1≥0,∀t2≥0.
Now, we can conclude that(51)DYt,s−DXt,s=∫t1+∞e−s1x1−t1LY2t∗s2G¯x1,t2G¯t1,t2−LX2t∗s2F¯x1,t2F¯t1,t2dx1≥∫t1+∞e−s1x1−t1LX2t∗s2G¯x1,t2G¯t1,t2−F¯x1,t2F¯t1,t2dx1≥0,for allt∈ℝ+2,for alls2≥0.
Thus, Theorem 1 together with Proposition2 provide the following result as a characterization property of the wfr order by bivariate Laplace transform ordering of residual lives. The proof is explicit, and thus we omit it.Theorem 2.
LetX⊤=X1,X2 and Y⊤=Y1,Y2 be two non-negative random pairs having joint s.f.s F¯ and G¯, respectively. Then,(52)X≤Ywfrif,and only if,Xt≤YBLtt,i=1,2,for allt∈ℝ+2.


## 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 thewfr order between lifetime pairs of X and Y by means of the wmrl order of typical transformations of X and Y. Descriptions of some bivariate aging notions in terms of the Laplace transform ordering of residual lives of X and a characterization result for a bivariate IFR aging property using a weaker bivariate DMRL aging behavior are given.Theorem 3.
LetX and Y be two lifetime random pairs with s.f.s F¯ and G¯, respectively. Suppose that T is a non-negative random pair which is independent of both X and Y. Then,(53)X≤Ywfr⟺X∧T≤Ywmrl∧T.Proof.
Denote byH¯ the joint s.f of T. Suppose that X∧T and Y∧T have respective joint s.f.s F¯∗ and G¯∗ which are given, from assumption, as(54)F¯∗x=H¯xF¯x,G¯∗x=H¯xG¯x,where x=x1,x2∈ℝ+2. Trivially, X≤Ywfr is equivalent to X∧T≤Ywfr∧T. Thus, X∧T≤Ywmrl∧T. Conversely, since for all non-negative random pairs T which are independent from X and Y, we have X∧T≤Ywmrl∧T, by considering Ts=Ts1,Ts2 where Ts1 and Ts2 are independent exponential random variables, one has X∧Ts≤Ywmrl∧Ts, for all si≥0i=1,2. From definition, this leads to the inequalities (55) and (56) given below being satisfied for all t,s∈ℝ+2 as(55)m1,X∧Tst=∫t1+∞e−s1x1−s2t2F¯x1,t2e−s1t1−s2t2F¯t1,t2dx1≤∫t1+∞e−s1x1−s2t2G¯x1,t2e−s1t1−s2t2G¯t1,t2dx1=m1,Y∧Tst,and(56)m2,X∧Tst=∫t2+∞e−s1t1−s2x2F¯t1,x2e−s1t1−s2t2F¯t1,t2dx2≤∫t2+∞e−s1t1−s2x2G¯t1,x2e−s1t1−s2t2G¯t1,t2dx2=m2,Y∧Tst.
It is evident that (55) is equivalent to LX1t∗s1≤LY1t∗s1 and also (56) is equivalent to LX2t∗s2≤LY2t∗s2 both of which hold for all t,s∈ℝ+2. Therefore, in the spirit of Theorem 1, it follows that X≤Ywfr. This completes the proof.
The result of Theorem1 can be used to characterize the wfr order between X and Y using the wmrl order when applying on gsX=1−e−s1X1,1−e−s2X2 and gsY=1−e−s1Y1,1−e−s2Y2 as two s−dependent random pairs on 0,1.Theorem 4.
LetX and Y be two lifetime random pairs with s.f.s F¯ and G¯, respectively. Then,(57)X≤Ywfr⟺gsX≤gwmrlsY,for alls∈ℝ+2.Proof.
We first prove thatX≤Ywfr implies gsX≤gwmrlsY for every non-negative s=s1,s2. If X≤Ywfr and if we denote gs1a=1−e−s1a,a>0, and gs2b=1−e−s2b,b>0, then since gs1a and gs2b are increasing in a and b, respectively, by Theorem 6.D.4. of Shaked and Shanthikumar [16], gsX≤gwfrsY, and therefore gsX≤gwmrlsY. To prove the converse part, note that gsX and gsY have respective s.f.s(58)F¯gsXu=F¯−1s1ln1−u1,−1s2ln1−u2,G¯gsYu=G¯−1s1ln1−u1,−1s2ln1−u2,in which u=u1,u2 and ui∈0,1,i=1,2. Therefore,(59)m1,gsXu=∫u11F¯−1/s1ln1−w1,−1/s2ln1−u2F¯−1/s1ln1−u1,−1/s2ln1−u2dw1=∫−1/s1ln1−u1+∞s1e−s1x1F¯x1,−1/s2ln1−u2F¯−1/s1ln1−u1,−1/s2ln1−u2dx1≤∫−1/s1ln1−u1+∞s1e−s1x1G¯x1,−1/s2ln1−u2G¯−1/s1ln1−u1,−1/s2ln1−u2dx1=m1,gsYu,for allui∈0,1,s∈ℝ+2,and moreover,(60)m2,gsXu=∫u21F¯−1/s1ln1−u1,−1/s2ln1−w2F¯−1/s1ln1−u1,−1/s2ln1−u2dw2=∫−1/s2ln1−u2+∞s2e−s2x2F¯−1/s1ln1−u1,x2F¯−1/s1ln1−u1,−1/s2ln1−u2dx2≤∫−1/s2ln1−u2+∞s2e−s2x2G¯−1/s1ln1−u1,x2G¯−1/s1ln1−u1,−1/s2ln1−u2dx2=m2,gsYu,for allui∈0,1,s∈ℝ+2.
By takingui=1−e−siti where ti≥0,i=1,2, (59) implies that for all s,t∈ℝ+2,(61)LX1t∗s1=1s1m1,gsX1−e−s1t1,1−e−s2t2≤LY1t∗s1=1s1m1,gsY1−e−s1t1,1−e−s2t2,and, in parallel, (60) concludes for all s,t∈ℝ+2 that(62)LX2t∗s2=1s2m2,gsX1−e−s1t1,1−e−s2t2≤LY2t∗s2=1s2m2,gsY1−e−s1t1,1−e−s2t2.
Hence, applying Theorem1, 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 thatX⊤=X1,X2 is a random pair of lifetimes which has joint s.f. F¯, joint failure rate function h≡h1,X,h2,X, and joint m.r.l. function m≡m1,X,m2,X. Then,(i)
X is said to have bivariate increasing failure rate (BIFR) aging property whenever F¯t1+a1,t2+a2/F¯t1,t2 is non-increasing in ti≥0i=1,2 for every a=a1,a2∈ℝ+2 or equivalently if hi,Xt1,t2≤hi,Xt1+a1,t2+a2, for all ti≥0 and ai≥0, i=1,2.(ii)
X is said to have bivariate decreasing mean residual lifetime (BDMRL) aging property provided that mi,Xt1,t2≥mi,Xt1+a1,t2+a2, for all ti≥0 and ai≥0, i=1,2.
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 thatX is a non-negative random pair. Then,(63)Xis BIFR⟺X∧Esis BDMRL for everys∈ℝ+2.Proof.
It can be readily seen thatX is BIFR if, and only if, Xa≤Xwfr for every a∈ℝ+2. This together with Theorem 3 concludes that X is BIFR if, and only if, Xa∧Es≤Xwmrl∧Es for all a,s∈ℝ+2. It is not hard to prove that(64)Xa∧EsandX∧Esaare identical in distribution.
Therefore,X is BIFR if, and only if, X∧Esa≤Xwmrl∧Es for all a,s∈ℝ+2 which, equivalently, holds whenever X∧Es is BDMRL for every s∈ℝ+2.


## 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 vectorsX⊤=X1,…,Xp and Y⊤=Y1,…,Yp of lifetimes with partially differentiable s.f.s F¯ and G¯, respectively. According to Hu et al. [11], it is said that X is smaller than Y in the weak multivariate failure rate order (denoted by X≤Ywfr) whenever G¯t/F¯t is non-decreasing in t∈t∈ℝ+p:G¯t>0. It is also said that X is smaller than Y in the multivariate Laplace transform order (denoted by X≤YMLt) if LXs=Ee−s⊤X≥LYs=Ee−s⊤Y, for all s∈ℝ+p (see, e.g., Shaked and Shanthikumar [16]). Let us consider Xt=X1−t1,…,Xp−tp|X1>t1,…,Xp>tp and Yt=Y1−t1,…,Yp−tp|Y1>t1,…,Yp>tp as the multivariate residual lifetime vectors associated with X and Y, respectively, where t=t1,…,tp is a vector of time points in ℝ+p. We can also consider the marginal conditional residual lifetimes Xit=Xi−ti|X1>t1,…,Xp>tp and Yit=Yi−ti|Y1>t1,…,Yp>tp for every i=1,…p. The result of Theorem 1 may be extended to the case where X≤Ywfr, in the multivariate case, is equivalent to Xit≤YLtit,i=1,…,p, for all t∈ℝ+p where p>2. To develop the result of Theorem 2, we can also study whether X≤Ywfr, in the multivariate setting, is equivalent to Xt≤YMLtt, for all t∈ℝ+p where p>2. 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.



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