Characterizations of the exponential distribution by the concept of residual life at random time
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 and be two lifetimes random variables with distribution functions and and density functions and , respectively. Denote by the residual life of at a random time (RLRT) with distribution function . The idle time of the server in a queuing system can be expressed as a RLRT. In theory of reliability, if is regarded as the total random life of a warm stand-by unit with an age of , RLRT then represents the actual working time of the stand-by unit, see Yue and Cao (2000) for more details. If and are mutually independent and have a common support, then
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 with distribution function is exponential if, and only if the mean residual life of at either or is equal to the mean of , for all integers in some set of consecutive integers where and are the lifetimes of a standby system and a -out-of- system, respectively, each composed of 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 in a Hilbert space is called complete if the only element of which is orthogonal to every is the null element, that is here 0 stands for the zero element of .
Note that denotes the inner product of . In the present paper, we consider the Hilbert space , whose inner product is given by where and are two real-valued square
Characterizations based on residual life at
Suppose that is a sequence of independent and identically distributed (i.i.d.) nonnegative random variables with pdf , distribution function and survival function . Additionally, suppose that the lifetime random variable is independent of this sequence. Let be the order statistics from the first members of the sequence of ’s. For , the density function of the th order statistic is given by
For each
Characterizations based on residual life at
This section includes another exponential characterization on the basis of a sequence of heterogenous gamma random variables. Let be a sequence of independent random variables having gamma distribution with respective shape parameters and a common scale parameter . The pdf of is given by , for and . We shall denote it by . Suppose that is the lifetime of a device. Denote by 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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