Characterization of MRL order of fail-safe systems with heterogeneous exponential components
Introduction
It is well known that order statistics play a very important role in statistical inference, life testing, reliability theory, and many other areas. Let denote the order statistics from random variables . Then, the -th order statistic is the lifetime of a -out-of- system, which is a very popular structure of redundancy in fault-tolerant systems that has been used extensively in industrial and military systems. In particular, and correspond to the lifetimes of parallel and series systems, respectively. Order statistics have been studied quite extensively in the case when the observations are independent and identically distributed (i.i.d.). However, in some practical situations, observations are non-i.i.d. Due to the complicated form of distributions of order statistics in the non-i.i.d. case, only limited results are found in the literature. Interested readers may refer to David and Nagaraja (2003) and Balakrishnan and Rao, 1998a, Balakrishnan and Rao, 1998b for comprehensive discussions on this topic, and the recent review article of Balakrishnan (2007) for an elaborate review of developments on the independent and non-identically distributed (i.ni.d.) case.
Because of its nice mathematical form and the unique memoryless property, the exponential distribution has been widely used in statistics, reliability, operations research, life testing, and some other applied fields. Readers may refer to Barlow and Proschan (1975) and Balakrishnan and Basu (1995) for an encyclopedic treatment to developments on the exponential distribution. Here, we will focus on the second order statistics, viz., the lifetimes of the -out-of- systems, which are commonly referred to as fail-safe systems; see Barlow and Proschan (1965). Pledger and Proschan (1971) were among the first to carry out a stochastic comparison of order statistics from non-i.i.d. exponential random variables with corresponding ones from i.i.d. exponential random variables. Since then, many researchers have studied this problem with many different goals and viewpoints; see, for example, Proschan and Sethuraman (1976), Kochar and Rojo (1996), Dykstra et al. (1997), Bon and Paˇltaˇnea, 1999, Bon and Paˇltaˇnea, 2006, Khaledi and Kochar, 2000, Khaledi and Kochar, 2002, Kochar and Xu (2007), Paˇltaˇnea (2008), and Zhao et al. (2009).
In the following, we first recall some notions on stochastic orders that are most pertinent to the main results developed in the subsequent sections. Throughout, the term increasing is used for monotone non-decreasing and similarly the term decreasing is used for monotone non-increasing. Definition 1.1 For two random variables and with their densities , and distribution functions , , respectively, let and denote their survival functions. As the ratios in the statements below are well defined: is said to be smaller than in the likelihood ratio order (denoted by ) if is increasing in . is said to be smaller than in the hazard rate order (denoted by ) if is increasing in . is said to be smaller than in the stochastic order (denoted by ) if . is said to be smaller than in the mean residual life order (denoted by ) if, for all ,
The hazard rate order in (ii) implies both the usual stochastic order in (iii) and the mean residual life order in (iv), but neither the usual stochastic order nor the mean residual life order implies the other. For a comprehensive discussion on stochastic orders, interested readers may refer to Müller and Stoyan (2002) and Shaked and Shanthikumar (2007).
Let be independent exponential random variables with having hazard rate , , and be a random sample from an exponential distribution with common hazard rate . Bon and Paˇltaˇnea (2006) then showed that for . Recently, Paˇltaˇnea (2008) improved this result partially for the special case as follows:andwhere . Zhao et al. (2009) obtained the corresponding characterization on the likelihood ratio order as follows:where , and
In a similar way, in this paper, the corresponding analogues based on the mean residual life order are established. More precisely, it is proved thatwhere and , andThus, the two results in (5), (6) form nice extensions of those in (1), (2) due to Paˇltaˇnea (2008) based on -order, and in (3), (4) due to Zhao et al. (2009) based on -order.
Section snippets
Preliminaries
In this section, we first recall some basic definitions and notions and then present some useful lemmas.
Main results
Theorem 3.1 Let be independent exponential random variables with respective hazard rates , and be an independent random sample from an exponential population with common hazard rate . Then, Proof Necessity Suppose . Kochar and Korwar (1996) pointed out that the first sample spacing is independent of the second sample spacing . Let be an exponential random variable with hazard rate , and be a mixture of exponential random
Some examples
In this section, we present some special examples in order to illustrate the performance of the results established in Section 3. Let be independent exponential random variables with respective hazard rates , and be an independent random sample from an exponential population with common hazard rate . Example 4.1 The case of two i.i.d. exponential samples is the simplest one. Suppose . In this case, it is easy to check that and
Acknowledgment
Authors would like to thank the referee for his/her insightful suggestions which led to a considerable improvement in the presentation and quality of this manuscript.
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