Comparing Lifetimes of Series and Parallel Systems with Heterogeneous Fréchet Components

Longxiang Fang,Yanqin Wang

doi:10.3390/sym9010010

Abstract

In this paper, we discuss stochastic comparisons of lifetimes of series and parallel systems with heterogeneous Fréchet components in terms of the usual stochastic order, reversed hazard rate order and likelihood ratio order. The partial results established here extend some well-known results in the literature of Gupta et al. Specifically, first, we generalize the result of Theorem 2 from the usual stochastic order to the reversed hazard rate order. Second, we generalize the result of Theorem 3 from the reversed hazard rate order to the likelihood ratio order. Last, we generalize the result of Theorem 4 from the hazard rate order to the likelihood ratio order when shape parameter 0 < α ≤ 1 .

Highlights

A random variable X is said to have a Fréchet distribution if its cumulative distribution function is F ( x ) = e−( x −μ −α θ ), x > μ, α > 0, θ > 0, where μ, θ and α are location, scale and shape parameters, respectively
Fréchet components with respect to the usual stochastic order, the reversed hazard rate order and the hazard rate order based on location and scale parameters of the Fréchet distributed components
First, the established result in Theorem 3 generalizes the result of Theorem 2 in Gupta et al [6] from the usual stochastic order to the reversed hazard rate order

Summary

Introduction

A random variable X is said to have a Fréchet distribution if its cumulative distribution function (cdf) is. ≤ Xn:n denote the order statistics corresponding to the random variables X1 , . In reliability theory, the lifetime of a k-out-of-n system is the (n − k + 1)th order statistic of a set of n random variables representing the component lifetimes. Stochastic comparisons of parallel and series systems with heterogeneous components have been studied by many authors. Gupta et al [6] have considered the stochastic comparisons between the lifetimes of parallel/series systems arising from independently distributed Fréchet components with respect to the usual stochastic. We first study the usual stochastic order comparison for the lifetimes of the parallel and series systems with independently Fréchet distributed components based on vector majorization of shape parameters but fixed location and scale parameters.

Preliminaries

Results

Conclusions