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

Shrahili, Mansour; Kayid, Mohamed

doi:https://doi.org/10.1155/2022/8151159

Journal of Function Spaces

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

Research Article | Open Access

Volume 2022 | Article ID 8151159 | https://doi.org/10.1155/2022/8151159

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

Mansour Shrahili¹and Mohamed Kayid¹

Academic Editor: Abdul Rauf Khan

Received21 May 2022

Revised29 Jun 2022

Accepted27 Jul 2022

Published10 Aug 2022

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.

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 Section 2, 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 that is a lifetime random variable with c.d.f. and s.f. . Let us take 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 until which it has been alive, is denoted by defined by for . The random variable , as a time-dependent conditional random variable, has s.f. for all . Being dependent on time, the stochastic residual lifetime process may be of some interest in different contexts (see, e.g., Chung [29] and Pekalp [30]). Gathering data in accordance with appearing 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

Define the truncated random variable according which the vitality function of is defined as where . 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 be another lifetime random variable with s.f. , the m.r.l. function and the v.f. . Then, to compare the residual live of relative to the residual live of in different times, the following measures can be set: where whenever and 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 and are equal in distribution iff , for all , for any . The random variables 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 and be two lifetime random variables with c.d.f’s and and probability density functions (p.d.f’s) and , respectively. Denote by , the residual life of at the random time with distribution functionis obtained. It has been shown that the idle time of the server in a queuing system can be stated as a residual life at random time (cf. Dewan and Khaledi [37]). In the context of reliability, when is the total random life of a warm stand by unit with a random age of , the random variable then stands as the actual working time of the stand by unit (cf. Yue and Cao [33]). If and are statistically independent having a common support, then

For ease of reference, before stating the main fundamental characterization properties, we provide some useful notions in functional analysis.

Definition 1. The sequence is complete in a Hilbert space if the only element of which is orthogonal to every is the null element, that is, where stands for the zero element of .

We recall that signifies an inner product of . In the present paper, we utilize the Hilbert space , whose inner product is determined by in which and are two real-valued square integrable functions defined on . Notice that if is a complete sequence in the Hilbert space , then with converges in provided that and the limit equal to . 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]). Let be an absolutely continuous function defined on with , and let its derivative satisfy a.e. on . Then, under the assumption the sequence is complete on iff the function is monotone on .

In the continuing part of this paper, we consider the sequence of random variables constituting independent and identically distributed (i.i.d.) nonnegative random variables with p.d.f. , c.d.f. , and s.f. The element in the sequence may denote the time of occurrence of a certain event. Suppose that denotes a random lifetime which is independent of the sequence , and therefore, where is an arbitrary nonnegative function and the equality in the event is not strict.

3. Reliability Measures on a Time Scale of Order Statistics

We assume that be the order statistics of the partial set . The p.d.f. of the th-order statistic , for is

Fix and consider an integer . Let us suppose that represents the time of occurrence of the th event among a series of events which are occurred consecutively at the ordered times . For instance, these times may denote the times at which external shocks are absorbed by a system. It is then realized that the residual life of at the random time denoted by , represents the residual lifetime of a device with lifetime after the time at which the th event occurs.

Let have s.f. . We assume throughout the paper that , which is used to denote the lifetime of a device, has a finite mean. For any it is also assumed that has a finite mean. It follows that is independent of . Hence, the s.f. of is derived as in which is a nonnegative random variable having p.d.f.

To derive a relative measure on random residual lives, suppose that is random variable denoting the lifetime of another device. Let have s.f. and m.r.l. function given by . Then, has s.f. where is a nonnegative random variable with p.d.f. (10). Let have vitality function given by . To obtain another relative measure by means of the vitality function, we derive where follows the p.d.f. given in (10). In the following, using several measures on evaluated at the random time , we present some characterization property. The notation is used to show equality in distribution.

Theorem 3. Let be a sequence of i.i.d. nonnegative absolutely continues random variables with s.f. Let be a random lifetime which is independent of , for all Let be another random lifetime. Then, (i) iff , for all for some where is the m.r.l. function of (ii) iff , for all for some where is the vitality function of (iii) iff , for all for some where is the the s.f. of

Proof. To prove (i), first note that has p.d.f. (10). Since , we can write 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 and have equal distributions, thus for all i.e., in view of (13), , for all for any given To prove the “if” part, we utilize the concept of complexness in functional analysis. It can be observed that, for a fixed and for all , Let us set . It is then plain to see , for all , if and only if, where Now, if we select in Lemma 2, it concludes that the sequence is complete on From completeness property, it follows from (15) that , for all , which means that , for all i.e., , for all . This is equivalent to saying that and are identical in distribution. We prove the assertion (ii) now. We assume that . We have where the expectations are with respect to which follows the p.d.f. (11). To prove the “only if” part, we suppose that and have identical distributions and, therefore, for all i.e., in spirit of (17), , for all for any given To establish the “if” part, we proceed as in the proof of assertion (i). It can be seen that, for a predetermined and for all , Take and observe that , for all , if and only if, where By considering in Lemma 2, it follows that the sequence is complete on according which Equation (19) implies that , for all , which means that , for all i.e., , for all . This further implies that and have the same distribution. We finally prove the assertion (iii). It is implied by the iterated expectation rule and also using the assumption that and are independent for all that The proof of the “only if” part is trivial from (21). To prove the “if” part, observe that for a fixed and for all , If we take we get , for all , if and only if, where By a choice of in Lemma 2, it is realized that the sequence is complete on by which Equation (23) guarantees that , for all . This means that , for all i.e., and are equally distributed. The proof of the theorem is complete.

In Theorem 3, the times have been scaled in the following order , in which is a fixed natural number and . Therefore, the result of Theorem 3 can be applied on the first-order statistic , so that if for all then and are equal in distribution, where

However, when the time is scaled in the order , the result of Theorem 3 is not applicable. Note that . In this modified setting, we build the characterization properties. Note that the probability for survival of after is where is a nonnegative random variable with the following p.d.f. (see Equation (10)):

In a similar manner, the probability for being greater than is

Next, we present the relevant characterization properties. The proof being similar to that of Theorem 3 has been shortened by omitting repeated steps.

Theorem 4. Let be a sequence of i.i.d. nonnegative absolutely continues random variables with s.f. Let be a random lifetime which is independent of , for all Let be another random lifetime. Then, (i) iff , for all where is the m.r.l. function of (ii) iff , for all where is the vitality function of (iii) iff , for all where is the s.f. of

Proof. We prove assertion (i). Notice note that has p.d.f. (27) and we assume that . It can be seen in a similar way as in the proof of Theorem 3 that The “only if” part is trivial. To prove the “if” part, we observe that By assumption , for all , which holds iff where Let us take in Lemma 2. We know that the sequence is complete on Hence, it follows from (31) that , for all , which further implies that , for all i.e., and are identical in their distribution. To establish the assertion (ii) we suppose that . Thus, analogously as in the proof of Theorem 3 (ii), we get The proof for the “only if” part is straightforward. To establish the “if” part, we write for all , From assumption , for all , if and only if, in which By choosing in Lemma 2, it is found that the sequence is complete on by which Equation (35) implies that , for all , which means that , for all i.e., and are identical in distribution. We now prove the assertion (iii). We have The proof of the “only if” part is obvious from (37). To prove the “if” part, observe that for all , By assumption, it holds that , for all which stands true iff where When we take , the Lemma 2 is applicable by which Equation (39) yields , for all . This means that , for all i.e., and have equal distributions and hence the proof.

4. Reliability Measures on a Time Scale of Record Values

The observation is called an upper record, if the value it takes exceeds that of all previous observations. Thus, is an upper record if for every . Analogously, the observation 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 , which are called upper record times and are defined by with probability 1 and, for , . Then, the upper record value sequence is considered. Similarly, the times at which the lower record values appear are the random variables , which are called lower record times defined by with probability 1 and, for , . Then, the lower record value sequence is considered. Then, the random variable , as the th upper record, has p.d.f. and the random variable , as the th lower record has p.d.f.

It may be of interest to evaluate and measure the excess amount of future records of observations. Let us consider the r.v. which denotes the lifetime of a unit which is independent of , and hence, it is also independent of and . The random variable measures the excess amount of observations on which are greater than the th upper record of ’s. The random variable measures the excess amount of observations on which are greater than the th lower record of ’s.

We give here other relative measures on random residual live. Let be the lifetime of a device to be compared with another device with lifetime . Then, has s.f. where is a nonnegative random variable with the following p.d.f.:

Correspondingly, the relative measure has s.f. where follows the p.d.f.

By an analogous method, the relative measure is greater than with probability and also the relative measure is greater than with probability

Now, other characterization properties based on random residual life after subsequent (upper and lower) records are given.

Theorem 5. Let be a sequence of i.i.d. nonnegative absolutely continues random variables with s.f. Let be a random lifetime which is independent of , for all Let be another random lifetime. Then, (i) iff , for all where is the m.r.l. function of (ii) iff , for all where is the vitality function of (iii) iff , for all where is the s.f. of

Proof. We give the proof of assertion (i). Note note that has p.d.f. (44) and assume that . As in the proof of Theorem 3, we have The “only if” part is trivial. To prove the “if” part, we observe that In view of assumption , for all , which is satisfied iff where We can take in Lemma 2 from which we realize that the sequence is complete on It thus follows from (51) that , for all , which provides that , for all i.e., and are identical in distribution. To give credit to the assertion (ii), we assume that . Therefore, similarly as in the proof of Theorem 3 (ii), we obtain The proof of the “only if” part is very simple. To prove the “if” part for all , we can write By assumption , for all if where It is found by Lemma 2 that the sequence is complete on from which Equation (55) gives , for all , that is , for all i.e., and are identical in distribution. To prove the assertion (iii), we write The proof of the “only if” part is plain to follow from (57). To prove the “if” part, note that for all , we have It holds by assumption that , for all which holds true iff where Equation (59) together with Lemma 2 provides that , for all . This concludes that , for all i.e., and are equally distributed, and hence, the proof is completed.

The following result will be similarly as Theorem 5 can be proved.

Theorem 6. Let be a sequence of i.i.d. nonnegative absolutely continues random variables with s.f. Let be a random lifetime which is independent of , for all Let be another random lifetime. Then, (i) iff , for all where is the m.r.l. function of (ii) iff , for all where is the m.r.l. function of where is the vitality function of (iii) iff , for all where is the s.f. of

It 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 can be any lifetime distribution. However, if 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 (Lemma 2) 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 and , further perspectives on the relative behaviour of a system with lifetime in comparison with another system in the same environment with lifetime may be produced. The relative measure is the scaled residual live of in which the reciprocal m.r.l. function of adjusts the scale. In contrast, the relative measure is the scaled right tail of the live of where the reciprocal vitality function of regulates the scale. The random quantities and are indeed scale-free measures. For example, the moments of and may be of some interest when and are equally distributed, as in this case (resp. ) is closely related with coefficient of variation of (resp. ) 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 and 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 as to contribute in the area of relative aging of systems and also further characterization properties will be sought.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by Researchers Supporting Project number (RSP2022R464), King Saud University, Riyadh, Saudi Arabia.

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Copyright © 2022 Mansour Shrahili and Mohamed Kayid. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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