# Characterizations of Lifetime Distributions Using Two Relative Reliability Measures

Authors: Ghadah Alomani; Mohamed Kayid
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, a general characterization property considering two new dynamic relative reliability measures is obtained. The new dynamic relative reliability measures are expressed as the ratio of hazard rates and as the ratio of reversed hazard rates. The measures are evaluated partially at some sequential random times following a specific distribution. We show that several particular statistics, as random times, fulfill that specific distribution, and thus, the result is applicable in the context of the specified random times. The results are applied to some examples to characterize the Weibull distribution and the inverse Weibull distribution.


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## Body

## 1. Introduction

The aging process of lifetime units is very important in various fields of science to quantify it by a mathematical theory. In survival analysis and in some medical problems, the construction of models is based on the aging process of life spans. In the context of reliability, the aging process of individuals or life span subjects is modeled by the residual life (RL) span variable, which includes the current age of the element [1–4]. The hazard rate (h.r.) function which is the density function divided to the survival function (s.f.) is a useful quantity for measuring the instantaneous risk of failure of units operating at different ages [5, 6]. This quantity has attracted the attention of many researchers in the field, who have presented several statistical models specifically for fitting right-censored data [7, 8].The inspection process, unlike the aging process that considers the future time of failure of objects, is concerned with determining when failure has already occurred. This process has also been useful to researchers in several areas, including reliability and risk. In these contexts, the inactivity time (IT) or past lifetime variable is used to quantify the inspection process and identify earlier times of failure [9–13]. A useful reliability measure that takes into account the risk of previous failures immediately prior to an inspection time is the revered hazard rate (r.h.r.) function [14–17]. The r.h.r. function which equals with density function divided to the distribution function has also been used in reliability analysis to propose new models that are particularly useful for modeling left-censored data [18–21].In probability theory, equivalent conditions for identical distributions have been always important tools for further purposes (see, e.g., [22–26]). The aim of the current paper is to present some characterizations using two quantities related to h.r. and r.h.r. functions. The results are obtained using the well-known theory of completeness in functional analysis.In what follows, in Section2, two new random relative reliability measures in the context of RL at random time (RLRT) and IT at random time (ITRT) are proposed. In Section 3, we give the main result including a general characterization property obtained using the introduced random relative h.r. and the random relative r.h.r. evaluated in some partial sequence of time. In Section 4, some typical random sequences of lifetimes which fulfill the derived characterization property are considered. In Section 5, it is shown that the obtained characterization property can be developed for characterization of particular distributions. In Section 6, we close the paper with further illustrations along with additional considerations to be studied in the future.


## 2. Relative Measures of Lifetime Distributions

Suppose thatX represents a nonnegative random variable (r.v.) with an absolutely continuous distribution function (d.f.) FX and probability density function (p.d.f.) fX. Then, the conditional r.v. Xt=X−tX>t for t:FXt<1 is called the residual lifetime of a subject (such as an electrical device) with random lifetime X after the time that subject has age t. Denote by F¯X the s.f. of X, defined by F¯Xt=1−FXt. Let us assume that X has h.r. function hX, given by
(1)hXt≔limΔ⟶0+1ΔPXt≤Δ=fXtF¯Xt,which is valid for all t≥0 for which FXt<1. For instance, if a system has not failed until the time t, then it is considered a used system of age t, and the r.v. Xt denotes the remaining lifetime it has. Thus, a fresh device with the s.f. F¯x and p.d.f. fXx when it is in use and has age t will have the updated s.f. and p.d.f.
(2)F¯Xtx≔F¯Xt+xF¯Xt,x≥0,fXtx≔fXt+xF¯Xt,x≥0,respectively. Suppose Y is another nonnegative r.v. and denote by FY, fY, F¯Y, and hY its d.f., p.d.f., s.f., and h.r. functions which are defined similarly as X.For the sake of obtaining a relative measure of aging ofX with respect to Y, consider the following r.v.
(3)RlX,Yt=hYtXt,t:maxFXt,FYt<1,which is the RL of X multiplied by the h.r. of Y. Now, the limiting probability for failure of a device with lifetime X right after the time t in comparison with the limiting probability for failure of another device with lifetime Y immediately after the time t (i.e., the h.r. of Y), the following measure can be proposed:
(4)hX,Yt≔limΔ⟶0+1ΔPRlX,Yt≤Δ=hXthYt.Note that (4) corresponds to the p.d.f. of RlX,Yt evaluated at the point 0. The relative measure (4) has been frequently applied in literature, specially for comparison of coherent systems and other lifetime events (see, e.g., [27–33]). The function defined in (4) can be viewed as a relative risk, and it plays a role in the evaluation of the Kullback-Leibler information (see, for instance, [34]). The monotonicity of hX,Yt with respect to t induces that the device, with lifetime X, ages faster (resp., slower) than another device with lifetime Y when hX,Yt is increasing (resp., decreasing) in t (see [35]).In many circumstances, the interest is in past events and, specifically, the time elapsed since a failure or death of a subject is important to be measured. The RL can then be considered in a reversed time scale (see, e.g., [9]). The conditional r.v. Xt=t−XX≤t for t:FXt>0 is considered the IT of a subject with random lifetime X at the time t at which the failure of the subject has been detected for the first time. We suppose that X has r.h.r. function
(5)h~Xt≔limΔ⟶0+1ΔPXt≤Δ=fXtFXt,which is well-defined for all t>0 for which FXt>0. For example, a system which is under periodically inspection at the time t observed to be failed. The r.v. Xt indicates the interval time between the failure time of a system and the time at which the inspector finds the system failed. The r.v. Xt has s.f. and p.d.f.
(6)F¯Xtx≔FXt−xFXt,x≥0,fXtx≔fXt−xFXt,x≥0,respectively. Let us denote by h~Y the r.h.r. function of Y. Based on the concept of IT, another measure of X relative to Y is
(7)Rl~X,Yt=h~YtXt,t:minFXt,FYt>0,which, indeed, is the IT of X multiplied by the r.h.r. of Y. Now, the following measure can be proposed:
(8)h~X,Yt≔limΔ⟶0+1ΔPRl~X,Yt≤Δ=h~Xth~Yt.Note that (8) corresponds to the p.d.f. of Rl~X,Yt evaluated at the point 0. The measure (8) has been considered a relative quantity for two lifetime distributions in literature. The monotonicity of h~X,Yt in terms of t concludes that the device, with lifetime X, is faster (slower) in decreasing r.h.r. (DRHR) property than another device with lifetime Y when h~X,Yt decreases (resp., increases) in t (see, e.g., [17, 36]). The monotonicity of the function defined in (8) has also been investigated in Proposition 5.1 of Di Crescenzo and Longobardi [37].The concept of RL and IT which are related to a given certain time has been developed to random time. The random timeT is assumed to be nonnegative. Let T have d.f. H (see, for instance, [38, 39]). Then, XT=X−TX>T is called the RL of X at T which, when T and X are independent, has s.f.
(9)F¯XTx=∫0+∞F¯Xt+xdHt∫0+∞F¯XtdHt,x≥0,and the associated p.d.f. is
(10)fXTx=∫0+∞fx+tdHt∫0+∞F¯XtdHt,x≥0.In contrast to the RLRT,XT=T−XX≤T is called the ITRT of T. If X and T are independent, then XT has s.f.
(11)F¯XTx=∫0+∞FXt−xdHt∫0+∞FXtdHt,with corresponding p.d.f.
(12)fXTx=∫0+∞ft−xdHt∫0+∞FXtdHt.To update the quantities (4) and (8) in terms of randomness of, respectively, the current age of a subject and the time of observation of failure of an item, we define their random counterparts. First,
(13)hX,YT=limΔ⟶0+1ΔPRlX,YT≤Δ=∫0+∞hXthYtdH∗t,where
(14)dH∗t=F¯XtdHt∫0+∞F¯XtdHt.Similarly,(15)h~X,YT=limΔ⟶0+1ΔPRl~X,YT≤Δ=∫0+∞h~Xth~YtdH∗∗t,in which
(16)dH∗∗t=FXtdHt∫0+∞FXtdHt.In the context of characterizations of distributions, one can see that since the h.r. function is a unique characteristic of the parent distribution, thus in view of (4), hX,Yt=1, for all t≥0 if X is equal in distribution with Y. Further, as the r.h.r. function is also a unique characteristic of the underlying distribution, therefore, in spirit of (8), h~X,Yt=1, for all t>0 if X and Y are equally distributed. In the sequel, we seek whether these properties can be developed for quantities (13) and (15). In the residual part of the paper, the terms “increasing” and “decreasing” mean “nondecreasing” and “nonincreasing”; thus, the monotonicity properties of functions are supposed to be nonstrict throughout the paper.


## 3. Main Characterization Properties

In this section, using a technical lemma to reach completeness property in functional analysis, two characterization properties will be given. We first remind the concept of completeness of a sequence of real function.Definition 1.
The sequenceδ1,δ2,⋯ in a given Hilbert space H is considered complete if the sole member in H having the orthogonality property with respect to every δm is the null member, in the way
(17)ψ•δm=0,∀m∈ℕ⇒ψ=0,where 0 indicates the zero member in H.The notation• represents the inner product of H. The Hilbert space L2a,b, across this paper, is assumed to have an inner product as
(18)f1•f2=∫abf1xf2xdx,where fi,i=1,2 is a real-valued square integrable function in the domain a,b. It is noticeable that if δ1,δ2,⋯ is a complete sequence in the Hilbert space H, then ∑cmδm where cm=f•δm converges in H whenever ∑m=1+∞cm2<+∞, and the limit becomes identical to f. Higgins [40] provided further detailed discussion regarding this area.Lemma 2 [41].
Letφ be a function on a,b which is absolutely continuous so that φaφb≥0, and suppose that φ′x≠0 almost everywhere on a,b. Then, under the assumption
(19)1v1+1v2+⋯=+∞,inwhich1≤v1<v2<⋯,the sequence φv1x,φv2x,⋯ is complete on a<x<b if φ is monotone on a,b.The special case wherevm=m, for m=1,2,⋯, fulfills the result of Lemma 2, because it is well-known that ∑m=1+∞1/m=+∞. Therefore, when φ is absolutely continuous and monotone, as a result, φx,φ2x,⋯ constitutes a complete sequence of functions.We consider a sequential family of distributions forT as the random time and apply it to update H∗ and H∗∗ in (13) and (15), respectively. Two characterization properties will then be secured.Let us suppose that, form∈ℕ, the random time T′m has p.d.f.
(20)fTm′t=wtφmt∫0+∞wtφmtdt,t≥0,where w· and φ· are two functions which wt≥0 and φt≥0 for all t≥0 so that 0<∫0+∞wtφmtdt<+∞. We shall denote the c.d.f. corresponding to T′m by FT′m. The random h.r. (13) and the random r.h.r. (13) with Tm′,m∈ℕ in place of T as a sequence of random times, two characterization relations, are obtained as follows:Theorem 3.
LetT′m be a sequence of r.v.s independent of X and Y. Then,
(i)
X and Y are equally distributed, if there exists a fixed positive integer m0 such that hX,YTm′=1, for all m=m0,m0+1,⋯ where T′m has p.d.f. (20) with a monotone function φ(ii)
X and Y are equally distributed, if there exists a fixed positive integer m0 such that h~X,YTm′=1, for all m=m0,m0+1,⋯ in which T′m has p.d.f. (20) with a function φ which is monotoneProof.
We first prove assertion (i). We assume thatX and Y are two nonnegative r.v.s with c.d.f.s F and G, and p.d.f.s f and g, respectively. By (13), we get
(21)hX,YTm′=EhXTm∗hYTm∗=∫0+∞hXthYtdFTm∗t=∫0+∞hXthYtwtF¯Xtφmt∫0+∞wtF¯Xtφmtdtdt,where Tm∗ has p.d.f
(22)fTm∗t=wtF¯Xtφmt∫0+∞wtF¯Xtφmtdt,t≥0.
It is straightforward that ifX and Y have equal distribution, then hX,YTm′=1, for all m=m0,m0+1,⋯. To prove the converse, note that hX,YTm′−1=0, for all m=m0,m0+1,⋯, if
(23)∫0+∞hXthYt−1wtF¯Xtφmt∫0+∞wtF¯Xtφmtdtdt=0,forallm=m0,m0+1,⋯,which holds, equivalently if, ∫0+∞ltφmtdt=0, for all m=1,2,⋯, that is l•φm=0, for all m=1,2,⋯ where lt=hXt/hYt−1wtφm0−1tF¯Xt. By Lemma 2, lt=0, for all t≥0, i.e., hXt=hYt, for all t≥0, i.e., X is equal in distribution with Y. We now prove assertion (ii). By (15), we obtain
(24)h~X,YTm′=Eh~XTm∗∗h~YTm∗∗=∫0+∞h~Xth~YtdFTm∗∗t=∫0+∞h~Xth~YtwtFXtφmt∫0+∞wtFXtφmtdtdt,where Tm∗∗ has p.d.f
(25)fTm∗∗t=wtFXtφmt∫0+∞wtFXtφmtdt,t≥0.
It is evident that ifX and Y have equal distribution, then h~X,YTm′=1, for all m=m0,m0+1,⋯. To prove the reversed implication, we have h~X,YTm′−1=0, for all m=m0,m0+1,⋯, if
(26)∫0+∞h~Xth~Yt−1wtFXtφmt∫0+∞wtFXtφmtdtdt=0,forallm=m0,m0+1,⋯,which is satisfied, equivalently, if ∫0+∞l∗tφmtdt=0, for all m=1,2,⋯, that is, l∗•φm=0, for all m=1,2,⋯ where l∗t=h~Xt/h~Yt−1wtφm0−1tF¯Xt. By applying Lemma 2, l∗t=0, for all t≥0, which concludes that h~Xt=h~Yt, for all t≥0, that is, X is equal in distribution with Y.Remark 4.
The result of Theorem3 remains valid if T′m has p.d.f.
(27)fTm′t=wtφsmt∫0+∞wtφsmtdt,t≥0,where s1,s2,⋯ are selected such that φs1,φs2,⋯ is a complete sequence of functions when φ is monotone due to Lemma 2. In this case, we see that in the context of Theorem 3 (i), one has
(28)hX,YTm′=EhXTm∗hYTm∗,where Tm∗ has p.d.f.
(29)fTm∗t=wtF¯Xtφsmt∫0+∞wtF¯Xtφsmtdt,t≥0.
In parallel, in Theorem3 (ii), we have
(30)h~X,YTm′=Eh~XTm∗∗h~YTm∗∗,in which Tm∗∗ is considered an r.v. with p.d.f.
(31)fTm∗∗t=wtFXtφsmt∫0+∞wtFXtφsmtdt,t≥0.Arnold and Villasenor [42] pointed out that characterizations are particularly of interest when they can be used to assess the conceivable of certain assumptions on distributions via suitable tests of hypotheses. Characterization properties of distributions can be, particularly, applied to build goodness-of-fit tests of distributions. By consideration of a proper random variable T′m in Theorem 3, the relation hX,YTm′=1, for all m=m0,m0+1,⋯, and also the relation h~X,YTm′=1, for all m=m0,m0+1,⋯, can be potential indices to construct tests for the hypothesis H0:XandY which areequal in distribution, versus H1:XandY which are note equal in distribution which is an appropriate alternative.


## 4. Fulfilling Random Sequences as Random Times

In this section, several situations where particular statistics may be adopted as random times are provided. In the context of random sampling from a population, many statistics can be considered.


### 4.1. Order Statistics from Homogenous Populations

In reliability engineering, the lifetime of a coherent system is represented in terms of consecutive order statistics (see [43]) arisen from the components lifetimes in the system, while the lifetime of a standby system is stated based on the partial sum of the component lifetimes in the system (see, for example, [44]). Suppose that T1,T2,⋯ is a sequence of independent and identically distributed (i.i.d.) nonnegative r.v.s with p.d.f. γ, c.d.f. H, and s.f. H¯. In the sequel, assume that the lifetime r.v. X is independent of Ti’s. Let T1:m≤T2:m≤⋯≤Tm:m be the order statistics from the first m elements of the sequence of T1,T2,⋯. In the context of Theorem 3, as i is fixed, the ith order statistic Ti:m can be considered as T′m where m=i,i+1,⋯, so that in the setting of Theorem 3, one can choose m0=i. It is known that T′m has p.d.f.
(32)fTm′t=m!i−1!m−i!Hi−1tH¯m−itγt,forallt≥0,which coincides with (20) if we take wt=Hi−1t/H¯itγt and φt=H¯t. Note that φ is a monotone decreasing function. Hence, Theorem 3 (i) is applicable and concludes that X and Y are equally distributed, if there exists an i∈ℕ such that hX,YTi:m=1, for all m=i,i+1,⋯. In view of (32), Tm∗ in the proof of Theorem 3 (i), for a fixed i=1,2,⋯ and m=i,i+1,⋯, has p.d.f.
(33)fTm∗t=F¯XtHi−1tH¯m−itγt∫0+∞F¯XtHi−1tH¯m−itγtdt,forallt≥0.In terms of (32), the p.d.f. of Tm∗∗ in the proof of Theorem 3 (ii), for a fixed i=1,2,⋯ and m=i,i+1,⋯, is
(34)fTm∗∗t=FXtHi−1tH¯m−itγt∫0+∞FXtHi−1tH¯m−itγtdt,forallt≥0.Theorem3 (ii) implies that X and Y are equally distributed, if there exists an i∈ℕ for which h~X,YTi:m=1, for all m=i,i+1,⋯.In spirit of (32), T′m≔Tm:m for m=1,2,⋯, has p.d.f.
(35)fTm′t=wtHmt∫0+∞wtHmtdt,where wt=γt/Ht and, thus, (20) is reached. The p.d.f. of Tm∗ in the proof of Theorem 3 (i) for m=1,2,⋯, is given here by
(36)fTm∗t=F¯XtHm−1tγt∫0+∞F¯XtHm−1tγtdt,forallt≥0.By consideringφt=Ht which increases in t, Theorem 3 (i) applies and concludes that X and Y are equally distributed, if hX,YTm:m=1, for all m=1,2,⋯. From (32), Tm∗∗ in the proof of Theorem 3 (ii), for m=1,2,⋯, has p.d.f.
(37)fTm∗∗t=FXtHm−1tγt∫0+∞FXtHm−1tγtdt,forallt≥0.In this case, from Theorem3 (ii) it concludes that X and Y are equally distributed, if h~X,YTm:m=1, for all m=1,2,⋯.


### 4.2. Order Statistics from Heterogenous Populations

In previous subsection, order statistics from i.i.d. r.v.s have been considered. In the context of the proportional hazard rate model (PHRM) and the proportional revered hazard rate model (PRHRM), the condition thatT1,T2,⋯ are identically distributed can be relaxed, respectively, when the smallest order statistic and the greatest order statistic are considered in these models as two coming random times. Let us assume that T1,T2,⋯ are nonnegative independent r.v.s and, furthermore, assume that they are independent of X such that Ti has s.f. H¯λi,i=1,2,⋯ where λ1≥1 and λi>0,i=2,3,⋯. The r.v. Ti follows the PHRM. Let us take sm=∑i=1mλi such that ∑m=1+∞sm−1=+∞ in which s1=λ1≥1. In light of (32) and also from (27), the p.d.f. of T′m≔T1:m for m=1,2,⋯, is
(38)fTm′t=wtH¯smt∫0+∞wtH¯smtdt,where wt=γt/H¯t. By (32), the p.d.f. of Tm∗ in the proof of Theorem 3 (i) for m=1,2,⋯ is replaced by the p.d.f. (29) which yields
(39)fTm∗t=F¯XtH¯sm−1tγt∫0+∞F¯XtH¯sm−1tγtdt,forallt≥0.Therefore, ifφt=H¯t which is a decreasing function, then Theorem 3 (i) concludes that X and Y are equally distributed, if hX,YT1:m=1, for all m=1,2,⋯. The p.d.f. of Tm∗∗ in the proof of Theorem 3 (ii), for m=1,2,⋯, is replaced by the p.d.f. (31) which gives
(40)fTm∗∗t=FXtH¯sm−1tγt∫0+∞FXtH¯sm−1tγtdt,forallt≥0.Thus, by assigningφt=Ht which is an increasing function, Theorem 3 (ii) presents that X and Y are equal in distribution, if h~X,YT1:m=1, for all m=1,2,⋯.We now discuss the case when the PRHRM is used. In this setting, assume thatT1,T2,⋯ are nonnegative independent r.v.s which are independent of X so that Ti has c.d.f. Hλi,i=1,2,⋯ where λ1≥1 and λi>0,i=2,3,⋯. We again take sm=∑i=1mλi with the requirement ∑m=1+∞sm−1=+∞ in which s1=λ1≥1. In terms of (32), the p.d.f of T′m≔Tm:m for m=1,2,⋯ is
(41)fTm′t=wtHsmt∫0+∞wtHsmtdt,where wt=γt/Ht, and this is a particular case for the p.d.f. (27) in Remark 4. The p.d.f. of Tm∗ in the proof of Theorem 3 (i) for m=1,2,⋯ can be replaced by the p.d.f. (29) implying that
(42)fTm∗t=F¯XtHsm−1tγt∫0+∞F¯XtHsm−1tγtdt,forallt≥0.Hence, ifφt=Ht which is an increasing function, then Theorem 3 (i) concludes that X and Y are equally distributed, if hX,YTm:m=1, for all m=1,2,⋯. The p.d.f. of Tm∗∗ in the proof of Theorem 3 (ii), for m=1,2,⋯, can be replaced by the p.d.f. (31) which gives
(43)fTm∗∗t=FXtHsm−1tγt∫0+∞FXtHsm−1tγtdt,forallt≥0.By making the choice ofφt=Ht which is increasing, Theorem 3 (ii) concludes that X and Y are equally distributed, if h~X,YTm:m=1, for all m=1,2,⋯.


### 4.3. Record Values

Now, we consider random times in the framework of record statistics. The epoch times associated with a nonhomogeneous Poisson process can be thought as the consecutive record values of a sequence of i.i.d. nonnegative r.v.s (see, for instance, [45]).


#### 4.3.1. Upper Records

The r.v.Ti upon its observation is an upper record, if the value it takes is greater than the corresponding value for previous observations. Hence, Tj is considered to be an upper record if Tj>Ti for every i<j. By realizing the amounts of consecutive records, a random sequence of times is produced at which the records appear. Let us denote the ith element of this sequence by Ui, considered to be the time at which the ith upper record is reached. The origin of time is considered U0 which is assumed to be zero with probability one and, for j≥1, Uj=mini:Ti>TUi−1. The upper records are then TUm:m=1,2,⋯. Since Ti’s are lifetime r.v.s, thus T0=0. The r.v. T′m≔TUm as the mth upper record follows the p.d.f.
(44)fTm′t=wt−logH¯tm∫0+∞wt−logH¯tmdt,where wt=γt which coincides with (20). The p.d.f. of Tm∗ in the proof of Theorem 3 (i) for m=1,2,⋯ is here
(45)fTm∗t=F¯Xt−logH¯tmγt∫0+∞F¯Xt−logH¯tmγtdt.Thus, ifφt=−logH¯t which is increasing in t, then Theorem 3 (i) is applicable and concludes that X and Y are equally distributed, if hX,YTUm=1, for all m=1,2,⋯. The p.d.f. of Tm∗∗ in the proof of Theorem 3 (ii), for m=1,2,⋯ is given here by
(46)fTm∗∗t=FXt−logH¯tmγt∫0+∞FXt−logH¯tmγtdt,forallt≥0.Theorem3 (ii) yields X and Y are equally distributed, if h~X,YTUm=1, for all m=1,2,⋯.


#### 4.3.2. Lower Records

In contrast to the upper records, the r.v.Ti upon its observation is a lower record, if its value is smaller than the corresponding value for previous observations. In such situation, Tj is considered to be a lower record if Tj<Ti for every i<j. Holding the amount of these records, a random sequence of times is produced at which the lower records occur. We denote the ith element of this sequence by Ri, considered to be the time at which the ith lower record is achieved. The origin of time is considered here to be R0 assumed to be zero and, for j≥1, Rj=mini:Ti<TUi−1. The lower records are recognized as TLm:m=1,2,⋯. It is known that T0=0. The r.v. T′m≔TLm as the mth lower record follows the p.d.f.
(47)fTm′t=wt−logHtm∫0+∞wt−logHtmdt,in which wt=γt, and this fulfills (20). The p.d.f. of Tm∗ in the proof of Theorem 3 (i) for m=1,2,⋯ is
(48)fTm∗t=F¯Xt−logHtmγt∫0+∞F¯Xt−logHtmγtdt.Hence, takingφt=−logHt which is decreasing in t, by Theorem 3 (i), we deduce that X and Y are equally distributed, if hX,YTLm=1, for all m=1,2,⋯. The p.d.f. of Tm∗∗ in the proof of Theorem 3 (ii), for m=1,2,⋯ is
(49)fTm∗∗t=FXt−logHtmγt∫0+∞FXt−logHtmγtdt,forallt≥0.Theorem3 (ii) provides that X and Y are equally distributed, if h~X,YTLm=1, for all m=1,2,⋯.


### 4.4. Convolution of Heterogenous Gamma Populations

Before closing this section, we apply the characterization property given in Theorem3 to partial sums of gamma r.v.s, with restricted shape parameters, as another fulfilling random sequence. In the next example, we show that a complete sequence is generated by the gamma distribution.Example 1.
SupposeTi has p.d.f. fTit=tαi−1βαie−βt/Γαi, for i=1,2,⋯, where αi>0 and β>0. Denote T′m≔T1+T2+⋯+Tm. It is a well-known result in probability that convolution of independent r.v.s. following gamma distribution again follows the gamma distribution. Hence, Tm′ has a gamma distribution with p.d.f.
(50)fTm′t=wttηm∫0+∞wttηm,where wt=e−βt/t and ηm=∑i=1mαi. If ∑m=1+∞ηm−1=+∞, whenever η1=α1≥1, then tηm,m=1,2,⋯ is a complete sequence due to Lemma 2.The p.d.f. ofTm∗ in the proof of Theorem 3 (i) for m=1,2,⋯ is
(51)fTm∗t=F¯Xttηm−1e−βt∫0+∞F¯Xttηm−1e−βtdt.The choiceφt=t as an increasing function together with the discussion in Example 1 will make Theorem 3 (i) applicable according which X and Y are equally distributed, if hX,YTm′=1, for all m=1,2,⋯. The p.d.f. of Tm∗∗ in the proof of Theorem 3 (ii), for m=1,2,⋯, is
(52)fTm∗∗t=FXttηm−1e−βt∫0+∞FXttηm−1e−βtdt.Thus, it is proved thatX and Y are equally distributed, if h~X,YTm′=1, for all m=1,2,⋯.


## 4.1. Order Statistics from Homogenous Populations

In reliability engineering, the lifetime of a coherent system is represented in terms of consecutive order statistics (see [43]) arisen from the components lifetimes in the system, while the lifetime of a standby system is stated based on the partial sum of the component lifetimes in the system (see, for example, [44]). Suppose that T1,T2,⋯ is a sequence of independent and identically distributed (i.i.d.) nonnegative r.v.s with p.d.f. γ, c.d.f. H, and s.f. H¯. In the sequel, assume that the lifetime r.v. X is independent of Ti’s. Let T1:m≤T2:m≤⋯≤Tm:m be the order statistics from the first m elements of the sequence of T1,T2,⋯. In the context of Theorem 3, as i is fixed, the ith order statistic Ti:m can be considered as T′m where m=i,i+1,⋯, so that in the setting of Theorem 3, one can choose m0=i. It is known that T′m has p.d.f.
(32)fTm′t=m!i−1!m−i!Hi−1tH¯m−itγt,forallt≥0,which coincides with (20) if we take wt=Hi−1t/H¯itγt and φt=H¯t. Note that φ is a monotone decreasing function. Hence, Theorem 3 (i) is applicable and concludes that X and Y are equally distributed, if there exists an i∈ℕ such that hX,YTi:m=1, for all m=i,i+1,⋯. In view of (32), Tm∗ in the proof of Theorem 3 (i), for a fixed i=1,2,⋯ and m=i,i+1,⋯, has p.d.f.
(33)fTm∗t=F¯XtHi−1tH¯m−itγt∫0+∞F¯XtHi−1tH¯m−itγtdt,forallt≥0.In terms of (32), the p.d.f. of Tm∗∗ in the proof of Theorem 3 (ii), for a fixed i=1,2,⋯ and m=i,i+1,⋯, is
(34)fTm∗∗t=FXtHi−1tH¯m−itγt∫0+∞FXtHi−1tH¯m−itγtdt,forallt≥0.Theorem3 (ii) implies that X and Y are equally distributed, if there exists an i∈ℕ for which h~X,YTi:m=1, for all m=i,i+1,⋯.In spirit of (32), T′m≔Tm:m for m=1,2,⋯, has p.d.f.
(35)fTm′t=wtHmt∫0+∞wtHmtdt,where wt=γt/Ht and, thus, (20) is reached. The p.d.f. of Tm∗ in the proof of Theorem 3 (i) for m=1,2,⋯, is given here by
(36)fTm∗t=F¯XtHm−1tγt∫0+∞F¯XtHm−1tγtdt,forallt≥0.By consideringφt=Ht which increases in t, Theorem 3 (i) applies and concludes that X and Y are equally distributed, if hX,YTm:m=1, for all m=1,2,⋯. From (32), Tm∗∗ in the proof of Theorem 3 (ii), for m=1,2,⋯, has p.d.f.
(37)fTm∗∗t=FXtHm−1tγt∫0+∞FXtHm−1tγtdt,forallt≥0.In this case, from Theorem3 (ii) it concludes that X and Y are equally distributed, if h~X,YTm:m=1, for all m=1,2,⋯.


## 4.2. Order Statistics from Heterogenous Populations

In previous subsection, order statistics from i.i.d. r.v.s have been considered. In the context of the proportional hazard rate model (PHRM) and the proportional revered hazard rate model (PRHRM), the condition thatT1,T2,⋯ are identically distributed can be relaxed, respectively, when the smallest order statistic and the greatest order statistic are considered in these models as two coming random times. Let us assume that T1,T2,⋯ are nonnegative independent r.v.s and, furthermore, assume that they are independent of X such that Ti has s.f. H¯λi,i=1,2,⋯ where λ1≥1 and λi>0,i=2,3,⋯. The r.v. Ti follows the PHRM. Let us take sm=∑i=1mλi such that ∑m=1+∞sm−1=+∞ in which s1=λ1≥1. In light of (32) and also from (27), the p.d.f. of T′m≔T1:m for m=1,2,⋯, is
(38)fTm′t=wtH¯smt∫0+∞wtH¯smtdt,where wt=γt/H¯t. By (32), the p.d.f. of Tm∗ in the proof of Theorem 3 (i) for m=1,2,⋯ is replaced by the p.d.f. (29) which yields
(39)fTm∗t=F¯XtH¯sm−1tγt∫0+∞F¯XtH¯sm−1tγtdt,forallt≥0.Therefore, ifφt=H¯t which is a decreasing function, then Theorem 3 (i) concludes that X and Y are equally distributed, if hX,YT1:m=1, for all m=1,2,⋯. The p.d.f. of Tm∗∗ in the proof of Theorem 3 (ii), for m=1,2,⋯, is replaced by the p.d.f. (31) which gives
(40)fTm∗∗t=FXtH¯sm−1tγt∫0+∞FXtH¯sm−1tγtdt,forallt≥0.Thus, by assigningφt=Ht which is an increasing function, Theorem 3 (ii) presents that X and Y are equal in distribution, if h~X,YT1:m=1, for all m=1,2,⋯.We now discuss the case when the PRHRM is used. In this setting, assume thatT1,T2,⋯ are nonnegative independent r.v.s which are independent of X so that Ti has c.d.f. Hλi,i=1,2,⋯ where λ1≥1 and λi>0,i=2,3,⋯. We again take sm=∑i=1mλi with the requirement ∑m=1+∞sm−1=+∞ in which s1=λ1≥1. In terms of (32), the p.d.f of T′m≔Tm:m for m=1,2,⋯ is
(41)fTm′t=wtHsmt∫0+∞wtHsmtdt,where wt=γt/Ht, and this is a particular case for the p.d.f. (27) in Remark 4. The p.d.f. of Tm∗ in the proof of Theorem 3 (i) for m=1,2,⋯ can be replaced by the p.d.f. (29) implying that
(42)fTm∗t=F¯XtHsm−1tγt∫0+∞F¯XtHsm−1tγtdt,forallt≥0.Hence, ifφt=Ht which is an increasing function, then Theorem 3 (i) concludes that X and Y are equally distributed, if hX,YTm:m=1, for all m=1,2,⋯. The p.d.f. of Tm∗∗ in the proof of Theorem 3 (ii), for m=1,2,⋯, can be replaced by the p.d.f. (31) which gives
(43)fTm∗∗t=FXtHsm−1tγt∫0+∞FXtHsm−1tγtdt,forallt≥0.By making the choice ofφt=Ht which is increasing, Theorem 3 (ii) concludes that X and Y are equally distributed, if h~X,YTm:m=1, for all m=1,2,⋯.


## 4.3. Record Values

Now, we consider random times in the framework of record statistics. The epoch times associated with a nonhomogeneous Poisson process can be thought as the consecutive record values of a sequence of i.i.d. nonnegative r.v.s (see, for instance, [45]).


### 4.3.1. Upper Records

The r.v.Ti upon its observation is an upper record, if the value it takes is greater than the corresponding value for previous observations. Hence, Tj is considered to be an upper record if Tj>Ti for every i<j. By realizing the amounts of consecutive records, a random sequence of times is produced at which the records appear. Let us denote the ith element of this sequence by Ui, considered to be the time at which the ith upper record is reached. The origin of time is considered U0 which is assumed to be zero with probability one and, for j≥1, Uj=mini:Ti>TUi−1. The upper records are then TUm:m=1,2,⋯. Since Ti’s are lifetime r.v.s, thus T0=0. The r.v. T′m≔TUm as the mth upper record follows the p.d.f.
(44)fTm′t=wt−logH¯tm∫0+∞wt−logH¯tmdt,where wt=γt which coincides with (20). The p.d.f. of Tm∗ in the proof of Theorem 3 (i) for m=1,2,⋯ is here
(45)fTm∗t=F¯Xt−logH¯tmγt∫0+∞F¯Xt−logH¯tmγtdt.Thus, ifφt=−logH¯t which is increasing in t, then Theorem 3 (i) is applicable and concludes that X and Y are equally distributed, if hX,YTUm=1, for all m=1,2,⋯. The p.d.f. of Tm∗∗ in the proof of Theorem 3 (ii), for m=1,2,⋯ is given here by
(46)fTm∗∗t=FXt−logH¯tmγt∫0+∞FXt−logH¯tmγtdt,forallt≥0.Theorem3 (ii) yields X and Y are equally distributed, if h~X,YTUm=1, for all m=1,2,⋯.


### 4.3.2. Lower Records

In contrast to the upper records, the r.v.Ti upon its observation is a lower record, if its value is smaller than the corresponding value for previous observations. In such situation, Tj is considered to be a lower record if Tj<Ti for every i<j. Holding the amount of these records, a random sequence of times is produced at which the lower records occur. We denote the ith element of this sequence by Ri, considered to be the time at which the ith lower record is achieved. The origin of time is considered here to be R0 assumed to be zero and, for j≥1, Rj=mini:Ti<TUi−1. The lower records are recognized as TLm:m=1,2,⋯. It is known that T0=0. The r.v. T′m≔TLm as the mth lower record follows the p.d.f.
(47)fTm′t=wt−logHtm∫0+∞wt−logHtmdt,in which wt=γt, and this fulfills (20). The p.d.f. of Tm∗ in the proof of Theorem 3 (i) for m=1,2,⋯ is
(48)fTm∗t=F¯Xt−logHtmγt∫0+∞F¯Xt−logHtmγtdt.Hence, takingφt=−logHt which is decreasing in t, by Theorem 3 (i), we deduce that X and Y are equally distributed, if hX,YTLm=1, for all m=1,2,⋯. The p.d.f. of Tm∗∗ in the proof of Theorem 3 (ii), for m=1,2,⋯ is
(49)fTm∗∗t=FXt−logHtmγt∫0+∞FXt−logHtmγtdt,forallt≥0.Theorem3 (ii) provides that X and Y are equally distributed, if h~X,YTLm=1, for all m=1,2,⋯.


## 4.3.1. Upper Records

The r.v.Ti upon its observation is an upper record, if the value it takes is greater than the corresponding value for previous observations. Hence, Tj is considered to be an upper record if Tj>Ti for every i<j. By realizing the amounts of consecutive records, a random sequence of times is produced at which the records appear. Let us denote the ith element of this sequence by Ui, considered to be the time at which the ith upper record is reached. The origin of time is considered U0 which is assumed to be zero with probability one and, for j≥1, Uj=mini:Ti>TUi−1. The upper records are then TUm:m=1,2,⋯. Since Ti’s are lifetime r.v.s, thus T0=0. The r.v. T′m≔TUm as the mth upper record follows the p.d.f.
(44)fTm′t=wt−logH¯tm∫0+∞wt−logH¯tmdt,where wt=γt which coincides with (20). The p.d.f. of Tm∗ in the proof of Theorem 3 (i) for m=1,2,⋯ is here
(45)fTm∗t=F¯Xt−logH¯tmγt∫0+∞F¯Xt−logH¯tmγtdt.Thus, ifφt=−logH¯t which is increasing in t, then Theorem 3 (i) is applicable and concludes that X and Y are equally distributed, if hX,YTUm=1, for all m=1,2,⋯. The p.d.f. of Tm∗∗ in the proof of Theorem 3 (ii), for m=1,2,⋯ is given here by
(46)fTm∗∗t=FXt−logH¯tmγt∫0+∞FXt−logH¯tmγtdt,forallt≥0.Theorem3 (ii) yields X and Y are equally distributed, if h~X,YTUm=1, for all m=1,2,⋯.


## 4.3.2. Lower Records

In contrast to the upper records, the r.v.Ti upon its observation is a lower record, if its value is smaller than the corresponding value for previous observations. In such situation, Tj is considered to be a lower record if Tj<Ti for every i<j. Holding the amount of these records, a random sequence of times is produced at which the lower records occur. We denote the ith element of this sequence by Ri, considered to be the time at which the ith lower record is achieved. The origin of time is considered here to be R0 assumed to be zero and, for j≥1, Rj=mini:Ti<TUi−1. The lower records are recognized as TLm:m=1,2,⋯. It is known that T0=0. The r.v. T′m≔TLm as the mth lower record follows the p.d.f.
(47)fTm′t=wt−logHtm∫0+∞wt−logHtmdt,in which wt=γt, and this fulfills (20). The p.d.f. of Tm∗ in the proof of Theorem 3 (i) for m=1,2,⋯ is
(48)fTm∗t=F¯Xt−logHtmγt∫0+∞F¯Xt−logHtmγtdt.Hence, takingφt=−logHt which is decreasing in t, by Theorem 3 (i), we deduce that X and Y are equally distributed, if hX,YTLm=1, for all m=1,2,⋯. The p.d.f. of Tm∗∗ in the proof of Theorem 3 (ii), for m=1,2,⋯ is
(49)fTm∗∗t=FXt−logHtmγt∫0+∞FXt−logHtmγtdt,forallt≥0.Theorem3 (ii) provides that X and Y are equally distributed, if h~X,YTLm=1, for all m=1,2,⋯.


## 4.4. Convolution of Heterogenous Gamma Populations

Before closing this section, we apply the characterization property given in Theorem3 to partial sums of gamma r.v.s, with restricted shape parameters, as another fulfilling random sequence. In the next example, we show that a complete sequence is generated by the gamma distribution.Example 1.
SupposeTi has p.d.f. fTit=tαi−1βαie−βt/Γαi, for i=1,2,⋯, where αi>0 and β>0. Denote T′m≔T1+T2+⋯+Tm. It is a well-known result in probability that convolution of independent r.v.s. following gamma distribution again follows the gamma distribution. Hence, Tm′ has a gamma distribution with p.d.f.
(50)fTm′t=wttηm∫0+∞wttηm,where wt=e−βt/t and ηm=∑i=1mαi. If ∑m=1+∞ηm−1=+∞, whenever η1=α1≥1, then tηm,m=1,2,⋯ is a complete sequence due to Lemma 2.The p.d.f. ofTm∗ in the proof of Theorem 3 (i) for m=1,2,⋯ is
(51)fTm∗t=F¯Xttηm−1e−βt∫0+∞F¯Xttηm−1e−βtdt.The choiceφt=t as an increasing function together with the discussion in Example 1 will make Theorem 3 (i) applicable according which X and Y are equally distributed, if hX,YTm′=1, for all m=1,2,⋯. The p.d.f. of Tm∗∗ in the proof of Theorem 3 (ii), for m=1,2,⋯, is
(52)fTm∗∗t=FXttηm−1e−βt∫0+∞FXttηm−1e−βtdt.Thus, it is proved thatX and Y are equally distributed, if h~X,YTm′=1, for all m=1,2,⋯.


## 5. Characterizations of Specific Distributions

In Section3 and Section 4, general characterizations of distributions have been derived using random relative (reversed) hazard rate measure when applied to some random sequences of time which are considered to be independent of the original random variables. In the context of Theorem 3 if either X or Y is fixed in a particular lifetime distribution, then a characterization of that specific distribution is produced. In this section, our aim is to characterize the Weibull distribution and the inverse Weibull distribution as two typical lifetime distributions.From Theorem3 (i), if there exist r.v.s T′1,T′2,⋯ so that T′m follows the p.d.f. fTm′t=wtφmt/∫0+∞wtφmtdt, such that hX,YTm′=1, for m=m0,m0+1,⋯ (for an m0∈ℕ), then X and Y are identical in distribution and vice versa. In view of Theorem 3 (i) and its proof if EhXTm∗/hYTm∗=1, for m=m0,m0+1,⋯ where Tm∗ has p.d.f. fTm∗t=wtF¯Xtφmt/∫0+∞wtF¯Xtφmtdt, then X and Y are equally distributed and vice versa. Example 2 and Example 3 are derived using Theorem 3 (i).Example 2.
SupposeY follows Weibull distribution with s.f. F¯Yt=exp−λtα,t≥0. Then, hYt=αλαtα−1, and if there exists an m0∈ℕ so that EhXTm∗/αλαTm∗α−1=1, for m=m0,m0+1,⋯, then X has Weibull distribution with parameters α and λ (we write X~Wα,λ) and vice versa. Equivalently, if there exist c0<1 and c1>0 such that ETm∗c0hXTm∗=c1, for all m=m0,m0+1,⋯ then X has Weibull distribution with α=1−c0 and λ=c1/1−c01−c0 and vice versa.Example 3.
SupposeX follows Wα,λ. If there exists an m0∈ℕ so that EαλαTm∗α−1/hYTm∗=1, for m=m0,m0+1,⋯ then Y~Wα,λ and vice versa. Equivalently, if there exist c0>−1 and c1>0 such that ETm∗c0/hYTm∗=c1, for all m=m0,m0+1,⋯, then Y has Weibull distribution with α=1+c0 and λ=1/c11+c01+c0 and vice versa.By Theorem3 (ii), it turns out that if there exist r.v.s T′1,T′2,⋯ so that T′m follows the p.d.f. fTm′t=wtφmt/∫0+∞wtφmtdt, such that h~X,YTm′=1, for m=m0,m0+1,⋯ (for an m0∈ℕ), then X and Y are identical in distribution and vice versa. This means from the proof of Theorem 3 (ii) that if Eh~XTm∗∗/h~YTm∗=1, for m=m0,m0+1,⋯ where Tm∗∗ has p.d.f. fTm∗∗t=wtFXtφmt/∫0+∞wtFXtφmtdt, then X and Y are equally distributed and vice versa. Example 4 and Example 5 are considered as applications of Theorem 3 (ii).Example 4.
Assume thatY follows inverse Weibull distribution with c.d.f. FYt=exp−1/λtα,t≥0. Then, h~Yt=α/λαtα+1, and if there exists an m0∈ℕ such that EλαTm∗∗α+1h~XTm∗∗/α=1, for m=m0,m0+1,⋯, then X has inverse Weibull distribution with the shape parameter α and scale parameter λ (we write X~IWα,λ) and vice versa. Equivalently, if there exist c0>1 and c1>0 such that ETm∗∗c0h~XT∗∗=c1, for all m=m0,m0+1,⋯ then X has inverse Weibull distribution with α=c0−1 and λ=c0−1/c1c0−1 and vice versa.Example 5.
Assume thatX follows IWα,λ. If there exists an m0∈ℕ so that Eα/λαTm∗∗α+1h~YTm∗∗=1, for m=m0,m0+1,⋯, then Y~IWα,λ and vice versa. Equivalently, if there exist c0>1 and c1>0 such that E1/Tm∗∗c0h~YTm∗∗=c1, for all m=m0,m0+1,⋯, then Y has inverse Weibull distribution with α=c0−1 and λ=c1c0−1c0−1 and vice versa.The characterizations of two standard families of lifetime distributions in the examples in this section can be developed to other (standard or nonstandard) families of lifetime distributions. Furthermore, in this context, considering specific sequences of random timesT′m as identified in Section 4 gives further characterizations of specific distributions. Let us suppose that X1,X2,⋯,Xm,Xm+1,⋯ are i.i.d. and that T′m=X1:m is the minimum order statistic among X1,X2,⋯,Xm. Then, X has a distribution equal with distribution of Y if there exists an m0∈ℕ such that
(53)EhXm+1X1:m+1hYX1:m+1=1,forallm=m0,m0+1,⋯.In parallel, if we considerT′m=Xm:m which is the maximum order statistic among X1,X2,⋯,Xm. Then, X has a distribution equal with distribution of Y, if there exists an m0∈ℕ such that
(54)Eh~Xm+1Xm+1:m+1h~YXm+1:m+1=1,forallm=m0,m0+1,⋯.


## 6. Conclusion

The hazard ratio and the reversed hazard ratio have been developed with random ages. Characterization relations using theses quantities, when evaluated convectively at some random times, have been presented. The general form of the distribution of sequence of random times has been satisfied by a number of particular sequences of random times including minimum and maximum order statistics, as well as moderate order statistics of i.i.d. random times from a general distribution, upper and lower record values of i.i.d. random times taken from a general distribution, minimum and maximum order statistics of independent but not identical random times from the PHRM and PRHM models, and partial sums of gamma distributed random times. In future studies, the generalized order statistics and the sequential order statistics are considered as possible random times to see whether such statistics when constitute a random sequence fulfill the main characterization property derived in this paper.The random times have been considered to be independent of the original random variable whose RLRT plays the main role. The new measures proposed in this paper can be developed to the case where the original random variable and the sequence of random time are dependent. This may arise in some situations. For example, assume that one considers a coherent system and wants to compare the random h.r. of one (specific) component working in system relative to another (specific) component at the random time at which the system fails. In this situation, it is apparent that the lifetime of system cannot be independent of the lifetime its components have. In the future, the possibility to study the dependency in the random relative h.r. measure as well as the random relative r.h.r. quantity of two lifetime units will be considered.



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