Elsevier

Statistics & Probability Letters

Volume 107, December 2015, Pages 164-169
Statistics & Probability Letters

Characterizations of the exponential distribution by the concept of residual life at random time

https://doi.org/10.1016/j.spl.2015.08.022Get rights and content

Abstract

In this paper, the convolution and the order statistics of k independent random lifetimes are considered as random times. Based on the concept of residual life at such random times, new characterizations of the exponential distribution are established.

Introduction

The concept of residual life at some random time plays an important role in reliability, survival analysis and life testing. In particular, preservation of aging classes and stochastic order relations have been studied in Yue and Cao (2000), Li and Zuo (2004), Misra et al. (2008), Cai and Zheng (2012) and Dewan and Khaledi (2014), among others.

Let X and Y be two lifetimes random variables with distribution functions F and G and density functions f and g, respectively. Denote by XY=d[XYX>Y] the residual life of X at a random time Y (RLRT) with distribution function FY. The idle time of the server in a GI/G/1 queuing system can be expressed as a RLRT. In theory of reliability, if X is regarded as the total random life of a warm stand-by unit with an age of Y, RLRT then represents the actual working time of the stand-by unit, see Yue and Cao (2000) for more details. If X and Y are mutually independent and have a common support, thenFY(x)=0[F(x+y)F(y)]g(y)dyP(X>Y).

Several characterization properties for the exponential distribution have been studied by many authors. Pfeifer (1982) established various characterizations by independent record increments. Gather (1988) obtained a new proof of a characterization by properties of order statistics. Riedel and Rossberg (1994) studied a characterization based on spacings of order statistics. Gharib (1996) obtained characterizations of the exponential distribution via mixing distributions. Some characterizations of other distributions have also been obtained in the literature (cf. Fashandi and Ahmadi, 2012 and the references therein). Recently, Arnold and Villasenor (2013) established characterizations based on order statistics of samples of size two.

In this paper, we present new exponential characterizations by properties of residual life at random time. In particular, we prove that a lifetime random variable X with distribution function F is exponential if, and only if the mean residual life of X at either Sk or Yi:k is equal to the mean of X, for all integers k in some set of consecutive integers where Sk and Yi:k are the lifetimes of a standby system and a (ki+1)-out-of-k system, respectively, each composed of k components with independent lifetimes.

The rest of the paper is organized as follows. First, we recall some notions of completeness in Section  2. Section  3 presents characterizations of the exponential distribution based on residual life at order statistics. Characterizations of the exponential distribution based on residual life at convolution of random variables are established in Section  4.

Section snippets

Preliminaries

For ease of reference, before stating exponential characterizations we provide some notions.

Definition 2.1

A sequence ϕ1,ϕ2, in a Hilbert space H is called complete if the only element of H which is orthogonal to every ϕn is the null element, that is f,ϕn=0,n1f=0, here 0 stands for the zero element of H.

Note that , denotes the inner product of H. In the present paper, we consider the Hilbert space L2[a,b], whose inner product is given by f,g=abf(x)g(x)dx, where f and g are two real-valued square

Characterizations based on residual life at Yi:k

Suppose that Y1,Y2, is a sequence of independent and identically distributed (i.i.d.) nonnegative random variables with pdf g, distribution function G and survival function Ḡ. Additionally, suppose that the lifetime random variable X is independent of this sequence. Let Y1:kY2:kYk:k be the order statistics from the first k members of the sequence of Yi’s. For i=1,2,,k, the density function of the ith order statistic Yi:k is given by gi:k(y)=i(ki)Gi1(y)Ḡki(y)g(y),for all  y>0.

For each

Characterizations based on residual life at Sk=i=1kYi

This section includes another exponential characterization on the basis of a sequence of heterogenous gamma random variables. Let Y1,Y2, be a sequence of independent random variables having gamma distribution with respective shape parameters α1,α2, and a common scale parameter β. The pdf of Yi,i=1,2, is given by g(y,αi,β)=yαi1βαieβy/Γ(αi), for y>0,αi>0 and β>0. We shall denote it by G(αi,β). Suppose that X is the lifetime of a device. Denote by Sk=i=1kYi the lifetime of a standby system

Acknowledgments

The authors are thankful to Co-Editor-in-Chief, an anonymous Associate Editor and a learned reviewer for their constructive comments on the earlier version of our paper. The second author sincerely thanks Dr. Nuria Torrado for her useful comments and suggestions regarding the first draft of this work. The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for funding this research group (No. RG-1435-036).

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