Characterization of MRL order of fail-safe systems with heterogeneous exponential components

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Abstract

Let X1,,Xn be independent exponential random variables with Xi having hazard rate λi, i=1,,n, and Y1,,Yn be another independent random sample from an exponential distribution with common hazard rate λ. The purpose of this paper is to examine the mean residual life order between the second order statistics X2:n and Y2:n from these two sets of variables. It is proved that X2:n is larger than Y2:n in terms of the mean residual life order if and only ifλ(2n-1)n(n-1)i=1n1Λi-n-1Λ,where Λ=i=1nλi and Λi=Λ-λi. It is also shown that X2:n is smaller than Y2:n in terms of the mean residual life order if and only ifλmin1inΛin-1.These results extend the corresponding ones based on hazard rate order and likelihood ratio order established by Paˇltaˇnea [2008. On the comparison in hazard rate ordering of fail-safe systems. Journal of Statistical Planning and Inference 138, 1993–1997] and Zhao et al. [2009. Likelihood ratio order of the second order statistic from independent heterogeneous exponential random variables. Journal of Multivariate Analysis 100, 952–962], respectively.

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 X1:nX2:nXn:n denote the order statistics from random variables X1,X2,,Xn. Then, the k-th order statistic Xk:n is the lifetime of a (n-k+1)-out-of-n 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, Xn:n and X1:n 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 (n-1)-out-of-n 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 X and Y with their densities f, g and distribution functions F, G, respectively, let F¯=1-F and G¯=1-G denote their survival functions. As the ratios in the statements below are well defined:

  • (i)

    X is said to be smaller than Y in the likelihood ratio order (denoted by XlrY) if g(x)/f(x) is increasing in x.

  • (ii)

    X is said to be smaller than Y in the hazard rate order (denoted by XhrY) if G¯(x)/F¯(x) is increasing in x.

  • (iii)

    X is said to be smaller than Y in the stochastic order (denoted by XstY) if G¯(x)F¯(x).

  • (iv)

    X is said to be smaller than Y in the mean residual life order (denoted by XmrlY) if, for all t0,

tF¯(x)dxF¯(t)tG¯(x)dxG¯(t).

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 X1,,Xn be independent exponential random variables with Xi having hazard rate λi, i=1,,n, and Y1,,Yn be a random sample from an exponential distribution with common hazard rate λ. Bon and Paˇltaˇnea (2006) then showed that Xk:nstYk:nλ1(nk)1i1<<iknλi1λik1/kfor 1kn. Recently, Paˇltaˇnea (2008) improved this result partially for the special case k=2 as follows:X2:nhrY2:nλλhr=1i<jnλiλjn2andX2:nhrY2:nλmin1inΛin-1,where Λi=j=1nλj-λi. Zhao et al. (2009) obtained the corresponding characterization on the likelihood ratio order as follows:X2:nlrY2:nλλlr=12n-12Λ(1)+Λ(3)-Λ(1)Λ(2)Λ2(1)-Λ(2),where Λ(k)=i=1nλik,k=1,2,3, andX2:nlrY2:nλi=1nλi-max1inλin-1.

In a similar way, in this paper, the corresponding analogues based on the mean residual life order are established. More precisely, it is proved thatX2:nmrlY2:nλλmrl=(2n-1)n(n-1)n=1n1Λi-n-1Λ,where Λ=i=1nλi and Λi=Λ-λi, andX2:nmrlY2:nλmin1inΛin-1.Thus, the two results in (5), (6) form nice extensions of those in (1), (2) due to Paˇltaˇnea (2008) based on hr-order, and in (3), (4) due to Zhao et al. (2009) based on lr-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 X1,,Xn be independent exponential random variables with respective hazard rates λ1,,λn, and Y1,,Yn be an independent random sample from an exponential population with common hazard rate λ. Then,X2:nmrlY2:nλλmrl=2n-1n(n-1)n=1n1Λi-n-1Λ.

Proof Necessity

Suppose λλmrl. Kochar and Korwar (1996) pointed out that the first sample spacing X1:n is independent of the second sample spacing X2:n-X1:n. Let T1 be an exponential random variable with hazard rate Λ, and T2 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 X1,,Xn be independent exponential random variables with respective hazard rates λ1,,λn, and Y1,,Yn be an independent random sample from an exponential population with common hazard rate μ.

Example 4.1

The case of two i.i.d. exponential samples is the simplest one. Suppose λ1==λn=λ. In this case, it is easy to check that λmrl=2n-1n(n-1)i=1n1Λi-n-1Λ=λand λ^=i=1nλi-max1i

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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