# Characterizations of Probability Distribution by Some Sequential Relative Reliability Measures: An Application of Completeness in a Hilbert Space

Authors: Mansour Shrahili; Mohamed Kayid
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, by means of the concept of completeness in functional analysis, we characterize probability distributions using several relative measures of residual life of an item at independent sequences of random times. These sequences are the order statistics and the record statistics of a random sequence of times. The reliability quantities of the mean residual life, the vitality function, and the survival function are considered to construct the main characteristics of distributions.


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## Body

## 1. Introduction

A characterization relation is a certain distributional or statistical property of a statistic or statistics that uniquely determines the associated probability distribution. The characterization properties derived in literature are mostly based on random samples from common univariate discrete and continuous distributions, and some multivariate continuous distributions are also considered. The main approach in such characterizations use the properties of sample moments, order statistics, record statistics, and reliability properties. The literature in the context of characterizations of probability distributions is extensive. We refer the reader to several review articles, monographs (e.g., Kagan et al. [1]), Rao and Shanbhag [2], Galambos and Kotz [3], Arnold et al. [4], and Galambos and Kotz [5]) and encyclopedic books on distributions by Johnson, Kotz, and their coauthors (Johnson et al. [6] and Kotz et al. [7]) have appeared to be excellent filters in the subject.There are some characterization properties of standard distributions using the properties of order statistics (see, e.g., Pfeifer [8], Bairamov and Ozkal [9], Huang and Su [10], Beg et al. [11], Kayid and Izadkhah [12], Akbari et al. [13], Hu and Lin [14], and Betsch and Ebner [15]). Furthermore, there are several characterization results of reputable probability distributions derived by the properties of record statistics (see, e.g., Witte [16], Kamps [17], Franco and Ruiz [18], Wesolowski and Ahsanullah [19], Nagaraja and Barlevy [20], Balakrishnan and Stepanov [21], Baratpour et al. [22], and Ahmadi [23]).There may be independent characteristics of a single distribution in different contexts so that when they are given, the underlying distribution of the associated random variable is characterized uniquely. For instance, in the context of reliability and survival analysis, the survival function (s.f.), the hazard rate (h.r.) function, the mean residual life (m.r.l.) function, and the vitality function (v.f.) determine the underlying distribution uniquely. In terms of these unique quantities, by holding particular relations involving them, one may also characterize a probability distribution (see, for example, Gupta and Keating [24], Navarro et al. [25], Nair and Sankaran [26], Szymkowiak [27], and Szymkowiak [28]).The rest of the paper is organized as follows. In Section2, we state some preliminary definitions and introduce some concepts in reliability theory and functional analysis. In Section 3, fundamental characterizations of distributions using weighted residual lives at random times which are order statistics of a partial subset of a random sequence of times are obtained. In Section 4, the characterization properties of distributions are developed using weighted residual lives at record values of the random sequence of times. Finally, in Section 5, we conclude the paper with further detailed remarks and outline the results that are of potential interest in future.


## 2. Preliminaries

Suppose thatX is a lifetime random variable with c.d.f. F and s.f. F¯≔1−F. Let us take X as the life length of a lifespan or any other item that has a lifetime. The residual lifetime of that lifespan or the item after time point t until which it has been alive, is denoted by Xt defined by Xt≔X−tX>t for t:Ft<1. The random variable Xt, as a time-dependent conditional random variable, has s.f. F¯tx=F¯t+x/F¯t, for all t,x≥0. Being dependent on time, the stochastic residual lifetime process Xt:t≥0 may be of some interest in different contexts (see, e.g., Chung [29] and Pekalp [30]). Gathering data in accordance with appearing Xt in time as it goes by provides a more insightful and also further informative method to infer about target population. The mean residual lifetime function is defined as
(1)mXt≔EXt=∫t+∞F¯xdxF¯t.Define the truncated random variableXt≔XX>t according which the vitality function of X is defined as
(2)vXt≔EXt=∫0+∞F¯x∨tdxF¯t,where x∨t=maxx,t. It is notable here and also it leads to some basic conclusions in the sequel that the mean residual lifetime function and the vitality function each is a characteristic of the underlying distribution (cf. Oakes and Dasu [31] and Ruiz and Navarro [32]). Let Y be another lifetime random variable with s.f. F¯Y, the m.r.l. function mY and the v.f. vY. Then, to compare the residual live of X relative to the residual live of Y in different times, the following measures can be set:
(3)R1X,Y;t≔X−tmYtX>t,R2X,Y;t≔XvYtX>t,R3X,Y;t≔Ix>tF¯Yt,where Ix>t=1 whenever x>t and Ix>t=0 elsewhere. Due to the fact that each of the m.r.l., v.f., and s.f. determines the parent distribution uniquely, we can readily observe that X and Y are equal in distribution iff ERiX,Y;t=1, for all t≥0, for any i=1,2,3. The random variables RiX,Y;t,i=1,2,3 can be considered a three relative stochastic process.The residual life at random time has been determined to be an important measure in reliability, survival analysis, and life testing (Yue and Cao [33], Li and Zuo [34], Misra et al.[35], Cai and Zheng [36], and Dewan and Khaledi [37]). Let X and T be two lifetime random variables with c.d.f’s F and FT and probability density functions (p.d.f’s) f and fT, respectively. Denote by XT=dX−TX>T, the residual life of X at the random time T with distribution functionFXTis obtained. It has been shown that the idle time of the server in a GI/G/1 queuing system can be stated as a residual life at random time (cf. Dewan and Khaledi [37]). In the context of reliability, when X is the total random life of a warm stand by unit with a random age of T, the random variable XT then stands as the actual working time of the stand by unit (cf. Yue and Cao [33]). If X and T are statistically independent having a common support, then
(4)FXTx=∫0+∞Fx+t−FtfTtdtPX>T.For ease of reference, before stating the main fundamental characterization properties, we provide some useful notions in functional analysis.Definition 1.
The sequenceψ1,ψ2, is complete in a Hilbert space H if the only element of H which is orthogonal to every ψn is the null element, that is,
(5)f,ψn=0,∀n≥1⇒f=0,where 0 stands for the zero element of H.We recall that·,· signifies an inner product of H. In the present paper, we utilize the Hilbert space L2a,b, whose inner product is determined by
(6)f1,f2=∫abf1xf2xdx,in which f1 and f2 are two real-valued square integrable functions defined on a,b. Notice that if ψ1,ψ2,⋯ is a complete sequence in the Hilbert space H, then ∑cnϕn with cn=f,ϕn converges in H provided that ∑cn2<∞, and the limit equal to f. See, for instance, Higgins [38] for more detailed discussions about the theory of completeness. The following result which is a key concept development shows the characterization results in this paper.Lemma 2 (see [39]).
Letψ be an absolutely continuous function defined on a,b with ψaψb≥0, and let its derivative satisfy ψ′x≠0 a.e. on a,b. Then, under the assumption
(7)∑k=1∞λk−1=∞,where1≤λ1<λ2<⋯,the sequence ψλ1,ψλ2,⋯ is complete on a,b iff the function ψ is monotone on a,b.In the continuing part of this paper, we consider the sequence of random variablesTn,n∈ℕ constituting independent and identically distributed (i.i.d.) nonnegative random variables with p.d.f. fT, c.d.f. FT, and s.f. F¯T. The element Ti,i=1,⋯,n in the sequence may denote the time of occurrence of a certain event. Suppose that X denotes a random lifetime which is independent of the sequence Tn,n∈ℕ, and therefore,
(8)PX>ψωT1,⋯,Tn,xωT1,⋯,Tn=t=PX>ψω,x,foralln=1,2,⋯,where ψ is an arbitrary nonnegative function and the equality in the event ωT1,⋯,Tn=ω is not strict.


## 3. Reliability Measures on a Time Scale of Order Statistics

We assume thatT1:n≤T2:n≤⋯≤Tn:n be the order statistics of the partial set T1,T2,⋯,Tn. The p.d.f. of the ith-order statistic Ti:n, for i=1,2,⋯n, is
(9)fTi:nt=n!i−1!n−i!FTi−1tF¯Tn−itfTt,forallt>0.Fixi∈ℕ and consider an integer n=i,i+1,⋯. Let us suppose that Ti:n represents the time of occurrence of the ith event among a series of n events which are occurred consecutively at the ordered times T1:n<T2:n<⋯<Tn:n. For instance, these times may denote the times at which n external shocks are absorbed by a system. It is then realized that the residual life of X at the random time Ti:n, denoted by XTi:n=X−Ti:nX>Ti:n, represents the residual lifetime of a device with lifetime X after the time at which the ith event occurs.LetX have s.f. F¯. We assume throughout the paper that X, which is used to denote the lifetime of a device, has a finite mean. For any n=i,i+1,⋯ it is also assumed that XTi:n has a finite mean. It follows that X is independent of Ti:n. Hence, the s.f. of XTi:n is derived as
(10)PXTi:n>x=PX>Ti:n+xPX>Ti:n=∫0+∞F¯t+xfTi:ntdt∫0+∞F¯tfTi:ntdt=∫0+∞F¯t+xF¯tF¯tfTi:nt∫0+∞F¯tfTi:ntdtdt=EF¯Ti,n∗+xF¯Ti,n∗,x≥0,in which Ti,n∗ is a nonnegative random variable having p.d.f.
(11)fTi,n∗t=F¯tfTi:nt∫0+∞F¯tfTi:ntdt=F¯tFTi−1tF¯Tn−itfTt∫0+∞F¯tFTi−1tF¯Tn−itfTtdt.To derive a relative measure on random residual lives, suppose thatY is random variable denoting the lifetime of another device. Let Y have s.f. F¯Y and m.r.l. function mY, given by mYt=∫t+∞F¯Yydy/F¯Yt. Then, XTi:n/mYTi:n has s.f.
(12)PXTi:nmYTi:n>x=PX−Ti:nmYTi:n>xX>Ti:n=PX>Ti:n+xmYTi:nPX>Ti:n=∫0+∞F¯t+xmYtfTi:ntdt∫0+∞F¯tfTi:ntdt=∫0+∞F¯t+xmYtF¯tF¯tfTi:nt∫0+∞F¯tfTi:ntdtdt=EF¯Ti,n∗+xmYTi,n∗F¯Ti,n∗,x≥0,where Ti,n∗ is a nonnegative random variable with p.d.f. (10). Let Y have vitality function vY, given by vYt=∫0+∞F¯Yy∨tdy/F¯Yt. To obtain another relative measure by means of the vitality function, we derive
(13)PXTi:nvYTi:n>x=PXvYTi:n>xX>Ti:n=PX>Ti:n∨xvYTi:nPX>Ti:n=∫0+∞F¯xvYt∨tfTi:ntdt∫0+∞F¯tfTi:ntdt=∫0+∞F¯xvYt∨tF¯tF¯tfTi:nt∫0+∞F¯tfTi:ntdtdt=EF¯xvYTi,n∗∨Ti,n∗F¯Ti,n∗,x≥0,where Ti,n∗ follows the p.d.f. given in (10). In the following, using several measures on X evaluated at the random time Ti:n, we present some characterization property. The notation =st is used to show equality in distribution.Theorem 3.
LetTn,n∈ℕ be a sequence of i.i.d. nonnegative absolutely continues random variables with s.f. F¯T. Let X be a random lifetime which is independent of Tn, for all n=1,2,⋯. Let Y be another random lifetime. Then,
(i)
X=stY iff EXTi:n/mYTi:n=1, for all n=i,i+1,⋯, for some i∈ℕ where mY is the m.r.l. function of Y(ii)
X=stY iff EXTi:n/vYTi:n=1, for all n=i,i+1,⋯, for some i∈ℕ where vY is the vitality function of Y(iii)
X=stY iff EIX>Ti:n/F¯YTi:n=1, for all n=i,i+1,⋯, for some i∈ℕ where F¯Y is the the s.f. of YProof.
To prove (i), first note thatTi,n∗ has p.d.f. (10). Since EXTi:n/mYTi:n<+∞, we can write
(14)EXTi:nmYTi:n=∫0+∞PXTi:nmYTi:n>xdx=∫0+∞EF¯Ti,n∗+xmYTi,n∗F¯Ti,n∗dx=E∫0+∞F¯Ti,n∗+xmYTi,n∗F¯Ti,n∗dx=E∫Ti,n∗+∞F¯xmYTi,n∗F¯Ti,n∗dx=EmXTi,n∗mYTi,n∗,where the change in the order of expectation and integral is due to the well-known Fubini theorem. To prove the “only if” part of the theorem, assume that X and Y have equal distributions, thus mXt=mYt, for all t≥0, i.e., in view of (13), EXTi:n/mYTi:n=1, for all n=i,i+1,⋯, for any given i∈ℕ. To prove the “if” part, we utilize the concept of complexness in functional analysis. It can be observed that, for a fixed i∈ℕ and for all n=i,i+1,⋯,
(15)EXTi:nmYTi:n−1=∫0+∞mXtmYt−1fTi,n∗tdt=∫0+∞mXtmYt−1F¯tFTi−1tF¯Tn−itfTt∫0+∞F¯tFTi−1tF¯Tn−itfTtdtdt.
Let us setm=n−i+1. It is then plain to see EXTi:n/mYTi:n−1=0, for all n∈i,i+1,⋯, if and only if,
(16)∫0+∞ϕtF¯Tmtdt=0,forallm∈ℕ,where
(17)ϕt=mXtmYt−1F¯tfTtFTi−1tF¯Tt,forallt≥0.
Now, if we selectψx=F¯Tx in Lemma 2, it concludes that the sequence F¯Tn:n∈ℕ is complete on ℝ+. From completeness property, it follows from (15) that ϕt=0, for all t≥0, which means that mXt/mYt−1=0, for all t≥0, i.e., mXt=mYt, for all t≥0. This is equivalent to saying that X and Y are identical in distribution. We prove the assertion (ii) now. We assume that EXTi:n/vYTi:n<+∞. We have
(18)EXTi:nvYTi:n=∫0+∞PXTi:nvYTi:n>xdx=∫0+∞EF¯xvYTi,n∗∨Ti,n∗F¯Ti,n∗dx=E∫0+∞F¯xvYTi,n∗∨Ti,n∗F¯Ti,n∗dx=E∫0+∞F¯x∨Ti,n∗vYTi,n∗F¯Ti,n∗dx=EvXTi,n∗vYTi,n∗,where the expectations are with respect to Ti,n∗ which follows the p.d.f. (11). To prove the “only if” part, we suppose that X and Y have identical distributions and, therefore, vXt=vYt, for all t≥0, i.e., in spirit of (17), EXTi:n/vYTi:n=1, for all n=i,i+1,⋯, for any given i∈ℕ. To establish the “if” part, we proceed as in the proof of assertion (i). It can be seen that, for a predetermined i∈ℕ and for all n=i,i+1,⋯,
(19)EXTi:nvYTi:n−1=∫0+∞vXtvYt−1fTi,n∗tdt=∫0+∞vXtvYt−1F¯tFTi−1tF¯Tn−itfTt∫0+∞F¯tFTi−1tF¯Tn−itfTtdtdt.
Takem=n−i+1 and observe that EXTi:n/vYTi:n−1=0, for all n∈i,i+1,⋯, if and only if,
(20)∫0+∞ϕ∗tF¯Tmtdt=0,forallm∈ℕ,where
(21)ϕ∗t=vXtvYt−1F¯tfTtFTi−1tF¯Tt,forallt≥0.
By consideringψx=F¯Tx in Lemma 2, it follows that the sequence F¯Tn:n∈ℕ is complete on ℝ+, according which Equation (19) implies that ϕ∗t=0, for all t≥0, which means that vXt/vYt−1=0, for all t≥0, i.e., vXt=vYt, for all t≥0. This further implies that X and Y have the same distribution. We finally prove the assertion (iii). It is implied by the iterated expectation rule and also using the assumption that X and Tn are independent for all n∈ℕ that
(22)EIX>Ti:nF¯YTi:n=EEIX>Ti:nF¯YTi:nTi:n=EF¯XTi:nF¯YTi:n.
The proof of the “only if” part is trivial from (21). To prove the “if” part, observe that for a fixed i∈ℕ and for all n=i,i+1,⋯,
(23)EIX>Ti:nF¯YTi:n−1=∫0+∞F¯XtF¯Yt−1fTi:ntdt=∫0+∞F¯XtF¯Yt−1FTi−1tF¯Tn−itfTt∫0+∞FTi−1tF¯Tn−itfTtdtdt.
If we takem=n−i+1 we get EIX>Ti:n/F¯YTi:n−1=0, for all n∈i,i+1,⋯, if and only if,
(24)∫0+∞ϕ∗∗tF¯Tmtdt=0,forallm∈ℕ,where
(25)ϕ∗∗t=F¯XtF¯Yt−1fTtFTi−1tF¯Tt,forallt≥0.
By a choice ofψx=F¯Tx in Lemma 2, it is realized that the sequence F¯Tn:n∈ℕ is complete on ℝ+, by which Equation (23) guarantees that ϕ∗∗t=0, for all t≥0. This means that F¯Xt/F¯Yt−1=0, for all t≥0, i.e., X and Y are equally distributed. The proof of the theorem is complete.In Theorem3, the times have been scaled in the following order Ti:i≤Ti:i+1≤⋯Ti:n≤Ti:n+1≤⋯, in which i is a fixed natural number and n=i,i+1,⋯. Therefore, the result of Theorem 3 can be applied on the first-order statistic T1:n, so that if ERiX,Y,T1:n=1 for all n=1,2,⋯ then X and Y are equal in distribution, where
(26)R1X,Y;T1:n=X−T1:nmYT1:nX>T1:n,R2X,Y;T1:n=XvYT1:nX>T1:n,R3X,Y;T1:n=IX>T1:nF¯YT1:n.However, when the time is scaled in the orderT1:1≤T2:2≤⋯≤Tn:n≤Tn+1:n+1≤⋯, the result of Theorem 3 is not applicable. Note that Tn:n=maxT1,T2,⋯,Tn. In this modified setting, we build the characterization properties. Note that the probability for survival of XTn:n/mYTn:n after x>0 is
(27)PXTn:nmYTn:n>x=EF¯Tn,n∗+xmYTn,n∗F¯Tn,n∗,where Tn,n∗ is a nonnegative random variable with the following p.d.f. (see Equation (10)):
(28)fTn,n∗t=F¯tFTn−1tfTt∫0+∞F¯tFTn−1tfTtdt.In a similar manner, the probability forXTn:n/vYTn:n being greater than x is
(29)PXTn:nvYTn:n>x=EF¯xvYTn,n∗∨Tn,n∗F¯Tn,n∗.Next, we present the relevant characterization properties. The proof being similar to that of Theorem3 has been shortened by omitting repeated steps.Theorem 4.
LetTn,n∈ℕ be a sequence of i.i.d. nonnegative absolutely continues random variables with s.f. F¯T. Let X be a random lifetime which is independent of Tn, for all n=1,2,⋯. Let Y be another random lifetime. Then,
(i)
X=stY iff EXTn:n/mYTn:n=1, for all n∈ℕ where mY is the m.r.l. function of Y(ii)
X=stY iff EXTn:n/vYTn:n=1, for all n∈ℕ where vY is the vitality function of Y(iii)
X=stY iff EIX>Tn:n/F¯YTn:n=1, for all n∈ℕ where F¯Y is the s.f. of YProof.
We prove assertion (i). Notice note thatTn,n∗ has p.d.f. (27) and we assume that EXTn:n/mYTn:n<+∞. It can be seen in a similar way as in the proof of Theorem 3 that
(30)EXTn:nmYTn:n=EmXTn,n∗mYTn,n∗,
The “only if” part is trivial. To prove the “if” part, we observe that(31)EXTn:nmYTn:n−1=∫0+∞mXtmYt−1fTn,n∗tdt=∫0+∞mXtmYt−1F¯tFTn−1tfTt∫0+∞F¯tFTn−1tfTtdtdt.
By assumptionEXTn:n/mYTn:n−1=0, for all n∈ℕ, which holds iff
(32)∫0+∞ϕ1tFTntdt=0,foralln∈ℕ,where
(33)ϕ1t=mXtmYt−1F¯tfTtFTt,forallt≥0.
Let us takeψx=FTx in Lemma 2. We know that the sequence FTn:n∈ℕ is complete on ℝ+. Hence, it follows from (31) that ϕ1t=0, for all t≥0, which further implies that mXt/mYt−1=0, for all t≥0, i.e., X and Y are identical in their distribution. To establish the assertion (ii) we suppose that EXTn:n/vYTn:n<+∞. Thus, analogously as in the proof of Theorem 3 (ii), we get
(34)EXTn:nvYTn:n=EvXTn,n∗vYTn,n∗.
The proof for the “only if” part is straightforward. To establish the “if” part, we write for alln=1,2,⋯,
(35)EXTn:nvYTn:n−1=∫0+∞vXtvYt−1fTn,n∗tdt=∫0+∞vXtvYt−1F¯tFTn−1tfTt∫0+∞F¯tFTn−1tfTtdtdt.
From assumptionEXTn:n/vYTn:n−1=0, for all n∈ℕ, if and only if,
(36)∫0+∞ϕ1∗tFTntdt=0,foralln∈ℕ,in which
(37)ϕ1∗t=vXtvYt−1F¯tfTtFTt,forallt≥0.
By choosingψx=FTx in Lemma 2, it is found that the sequence FTn:n∈ℕ is complete on ℝ+, by which Equation (35) implies that ϕ1∗t=0, for all t≥0, which means that vXt/vYt−1=0, for all t≥0, i.e., X and Y are identical in distribution. We now prove the assertion (iii). We have
(38)EIX>Tn:nF¯YTn:n=EF¯XTn:nF¯YTn:n.
The proof of the “only if” part is obvious from (37). To prove the “if” part, observe that for all n=1,2,⋯,
(39)EIX>Tn:nF¯YTn:n−1=∫0+∞F¯XtF¯Yt−1fTn:ntdt=∫0+∞F¯XtF¯Yt−1FTn−1tfTt∫0+∞FTi−1tfTtdtdt.
By assumption, it holds thatEIX>Tn:n/F¯YTn:n−1=0, for all n∈ℕ which stands true iff
(40)∫0+∞ϕ1∗∗tF¯Tntdt=0,foralln∈ℕ,where
(41)ϕ1∗∗t=F¯XtF¯Yt−1fTtFTt,forallt≥0.
When we takeψx=F¯Tx, the Lemma 2 is applicable by which Equation (39) yields ϕ1∗∗t=0, for all t≥0. This means that F¯Xt/F¯Yt−1=0, for all t≥0, i.e., X and Y have equal distributions and hence the proof.


## 4. Reliability Measures on a Time Scale of Record Values

The observationTi is called an upper record, if the value it takes exceeds that of all previous observations. Thus, Tj is an upper record if Tj>Ti for every i<j. Analogously, the observation Ti is called a lower record, if the value it takes falls behind all previous observations. The times at which the upper record values appear are given by the random variables Ui, which are called upper record times and are defined by U0 with probability 1 and, for j≥1, Uj=mini:Ti>TUi−1. Then, the upper record value sequence TUn:n=0,1,2,⋯ is considered. Similarly, the times at which the lower record values appear are the random variables Li, which are called lower record times defined by L0 with probability 1 and, for j≥1, Lj=mini:Ti<TLi−1. Then, the lower record value sequence TLn:n=0,1,2,⋯ is considered. Then, the random variable TUn, as the nth upper record, has p.d.f.
(42)fTUnt=−logF¯Ttnn!fTt,and the random variable TLn, as the nth lower record has p.d.f.
(43)fTLnt=−logFTtnn!fTt.It may be of interest to evaluate and measure the excess amount of future records of observations. Let us consider the r.v.X which denotes the lifetime of a unit which is independent of Tn,n=1,2,⋯, and hence, it is also independent of TUn and TLn. The random variable XTUn=X−TUnX>TUn measures the excess amount of observations on X which are greater than the nth upper record of Ti’s. The random variable XTLn=X−TLnX>TLn measures the excess amount of observations on X which are greater than the nth lower record of Ti’s.We give here other relative measures on random residual live. LetY be the lifetime of a device to be compared with another device with lifetime X. Then, XTUn/mYTUn has s.f.
(44)PXTUnmYTUn>x=PX−TUnmYTUn>xX>TUn=PX>TUn+xmYTUnPX>TUn=∫0+∞F¯t+xmYtfTUntdt∫0+∞F¯tfTUntdt=∫0+∞F¯t+xmYtF¯tF¯tfTUnt∫0+∞F¯tfTUntdtdt=EF¯Tn,n∗∗+xmYTn,n∗∗F¯Tn,n∗∗,where Tn,n∗∗ is a nonnegative random variable with the following p.d.f.:
(45)fTn,n∗∗t=F¯tfTUnt∫0+∞F¯tfTUntdt=F¯t−logF¯TtnfTt∫0+∞F¯t−logF¯TtnfTtdt.Correspondingly, the relative measureXTLn/mYTLn has s.f.
(46)PXTLnmYTLn>x=EF¯Tn,n∗∗∗+xmYTn,n∗∗∗F¯Tn,n∗∗∗,where Tn,n∗∗∗ follows the p.d.f.
(47)fTn,n∗∗∗t=F¯t−logFTtnfTt∫0+∞F¯t−logFTtnfTtdt.By an analogous method, the relative measureXTUn/vYTUn is greater than x with probability
(48)PXTUnvYTUn>x=EF¯xvYTn,n∗∗∨Tn,n∗∗F¯Tn,n∗∗,and also the relative measure XTLn/vYTLn is greater than x with probability
(49)PXTLnvYTLn>x=EF¯xvYTn,n∗∗∗∨Tn,n∗∗∗F¯Tn,n∗∗∗.Now, other characterization properties based on random residual life after subsequent (upper and lower) records are given.Theorem 5.
LetTn,n∈ℕ be a sequence of i.i.d. nonnegative absolutely continues random variables with s.f. F¯T. Let X be a random lifetime which is independent of Tn, for all n=1,2,⋯. Let Y be another random lifetime. Then,
(i)
X=stY iff EXTUn/mYTUn=1, for all n∈ℕ where mY is the m.r.l. function of Y(ii)
X=stY iff EXTUn/vYTUn=1, for all n∈ℕ where vY is the vitality function of Y(iii)
X=stY iff EIX>TUn/F¯YTUn=1, for all n∈ℕ where F¯Y is the s.f. of YProof.
We give the proof of assertion (i). Note note thatTn,n∗∗ has p.d.f. (44) and assume that EXTUn/mYTUn<+∞. As in the proof of Theorem 3, we have
(50)EXTUnmYTUn=EmXTn,n∗∗mYTn,n∗∗,
The “only if” part is trivial. To prove the “if” part, we observe that(51)EXTUnmYTUn−1=∫0+∞mXtmYt−1fTn,n∗∗tdt=∫0+∞mXtmYt−1F¯t−logF¯TtnfTt∫0+∞F¯t−logF¯TtnfTtdtdt.
In view of assumptionEXTUn/mYTUn−1=0, for all n∈ℕ, which is satisfied iff
(52)∫0+∞ϕ2t−logF¯Ttndt=0,foralln∈ℕ,where
(53)ϕ2t=mXtmYt−1F¯tfTt,forallt≥0.
We can takeψx=−logF¯Tx in Lemma 2 from which we realize that the sequence −logF¯Ttn:n∈ℕ is complete on ℝ+. It thus follows from (51) that ϕ2t=0, for all t≥0, which provides that mXt/mYt−1=0, for all t≥0, i.e., X and Y are identical in distribution. To give credit to the assertion (ii), we assume that EXTUn/vYTUn<+∞. Therefore, similarly as in the proof of Theorem 3 (ii), we obtain
(54)EXTUnvYTUn=EvXTn,n∗∗vYTn,n∗∗.
The proof of the “only if” part is very simple. To prove the “if” part for alln=1,2,⋯, we can write
(55)EXTUnvYTUn−1=∫0+∞vXtvYt−1fTn,n∗∗tdt=∫0+∞vXtvYt−1F¯t−logF¯TtnfTt∫0+∞F¯t−logF¯TtnfTtdtdt.
By assumptionEXTUn/vYTUn−1=0, for all n∈ℕ if
(56)∫0+∞ϕ2∗t−logF¯Ttndt=0,foralln∈ℕ,where
(57)ϕ1∗t=vXtvYt−1F¯tfTt,forallt≥0.
It is found by Lemma2 that the sequence −logF¯Ttn:n∈ℕ is complete on ℝ+, from which Equation (55) gives ϕ2∗t=0, for all t≥0, that is vXt/vYt−1=0, for all t≥0, i.e., X and Y are identical in distribution. To prove the assertion (iii), we write
(58)EIX>TUnF¯YTUn=EF¯XTUnF¯YTUn.
The proof of the “only if” part is plain to follow from (57). To prove the “if” part, note that for all n=1,2,⋯, we have
(59)EIX>TUnF¯YTUn−1=∫0+∞F¯XtF¯Yt−1fTUntdt=∫0+∞F¯XtF¯Yt−1−logF¯Ttnn!fTtdt.
It holds by assumption thatEIX>TUn/F¯YTUn−1=0, for all n∈ℕ which holds true iff
(60)∫0+∞ϕ2∗∗t−logF¯Ttndt=0,foralln∈ℕ,where
(61)ϕ2∗∗t=F¯XtF¯Yt−1fTt,forallt≥0.
Equation (59) together with Lemma 2 provides that ϕ2∗∗t=0, for all t≥0. This concludes that F¯Xt/F¯Yt−1=0, for all t≥0, i.e., X and Y are equally distributed, and hence, the proof is completed.The following result will be similarly as Theorem5 can be proved.Theorem 6.
LetTn,n∈ℕ be a sequence of i.i.d. nonnegative absolutely continues random variables with s.f. F¯T. Let X be a random lifetime which is independent of Tn, for all n=1,2,⋯. Let Y be another random lifetime. Then,
(i)
X=stY iff EXTLn/mYTLn=1, for all n∈ℕ where mY is the m.r.l. function of Y(ii)
X=stY iff EXTLn/vYTLn=1, for all n∈ℕ where mY is the m.r.l. function of Y where vY is the vitality function of Y(iii)
X=stY iff EIX>TLn/F¯YTLn=1, for all n∈ℕ where F¯Y is the s.f. of YIt must be mentioned here that the results obtained in this work differ from the characterization properties in Kayid and Izadkhah [12]. In fact, the characterization properties in this work represent a further development of the results obtained by Kayid and Izadkhah [12], since in our approach the distribution of Y can be any lifetime distribution. However, if Y is chosen to be an exponential r.v., then our characterization results in Theorems 3(i), 4(i), and 5(i) reduce to Theorems 3.1 and 3.2 in Kayid and Izadkhah [12], respectively.


## 5. Conclusion

In this paper, we have studied several characterization problems using the residual lifetime of an original item relative to the mean residual lifetime, the vitality function, or the survival function of another item. The ages at which the residual lifetimes applied have been considered to be independent random times as they are the order statistics of a partial set of a sequence of random lifetimes or the record statistics arising from the infinite version of the sequence of random lifetimes. The main idea for proving the results has been a technical Lemma (Lemma2) which is a basic result in functional analysis. The random residual lives are applicable to model the remaining useful life of high-quality products with respect to frailer products. Further strategies and descriptions using the introduced concept in reliability theory and survival analysis could be given.Recently, many researchers have focused on the relating aging properties of two lifespan by the monotonicity of ratio of their associated reliability quantities (see, e.g., Finkelstein [40], Kayid et al. [41], and He and Xie [42]). The authors believes that using the measures R1X,Y;t=X−t/mXtX>t and R2X,Y;t=X/vXtX>t, further perspectives on the relative behaviour of a system with lifetime X in comparison with another system in the same environment with lifetime Y may be produced. The relative measure R1X,Y;t is the scaled residual live of X in which the reciprocal m.r.l. function of Y adjusts the scale. In contrast, the relative measure R2X,Y;t is the scaled right tail of the live of X where the reciprocal vitality function of Y regulates the scale. The random quantities R1 and R2 are indeed scale-free measures. For example, the moments of R1 and R2 may be of some interest when X and Y are equally distributed, as in this case ER12X,Y;t=EXt2/E2Xt (resp. ER22X,Y;t=EXt2/E2Xt) is closely related with coefficient of variation of Xt (resp. Xt) which is a basic tool in the study of aging phenomenon or the vitality of the systems. However, extensions to the case where the age is random makes the study more dynamic and further the conclusions become quite broader. In situations where R1 and R2 are regarded as two stochastic processes, observation of the process will be sequentially in some random time not continually with time. In fact, it has been widely acknowledged that a stochastic process is partially observable. For example, in degradation models in which the lifetime is considered to be the first-passage time of some stochastic process (see, for example, Bordes et al. [43]). Therefore, the random relative residual lives and their averages measured in some sequences of random time such as order statics or record values may be practically beneficial. There are various situations where the times of occurrence of events are record values of a sequence or order statistics of a sample. For example, in the context of reliability engineering, a coherent system fails with the the failure of the consecutive order statistics of the lifetime of its components. The residual lifetime aftershocks arrived to a system may be an important quantity where the shocks which make further degradation in a highly reliable system are the ones which are record values.In the future of this study, the problem of characterizations of distributions in the cases when the lifetime depends on random ages will be studied. The question may be whether the current characterization properties remain valid and hold at the disposal of dependencies between lifetime and random ages. Constructing similar relative measures asRi,i=1,2 to contribute in the area of relative aging of systems and also further characterization properties will be sought.



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