Comparison of order statistics in a random sequence to the same statistics with i.i.d. variables

¹ Polytech-Lille, USTL, Laboratoire CNRS Painlevé, 59655 Villeneuve d'Ascq, France; jean-louis.bon@polytech-lille.fr
² Transilvania University of Braşov, Faculty of Mathematics and Computer Sciences, România.

Received: 14 December 2004
Revised: 25 May 2005

Abstract

The paper is motivated by the stochastic comparison of the reliability of non-repairable k-out-of-n systems. The lifetime of such a system with nonidentical components is compared with the lifetime of a system with identical components. Formally the problem is as follows. Let U_i,i = 1,...,n, be positive independent random variables with common distribution F. For λ_i > 0 and µ > 0, let consider X_i = U_i/λ_i and Y_i = U_i/µ, i = 1,...,n. Remark that this is no more than a change of scale for each term. For k ∈ {1,2,...,n}, let us define X_k:n to be the kth order statistics of the random variables X₁,...,X_n, and similarly Y_k:n to be the kth order statistics of Y₁,...,Y_n. If X_i,i = 1,...,n, are the lifetimes of the components of a n+1-k-out-of-n non-repairable system, then X_k:n is the lifetime of the system. In this paper, we give for a fixed k a sufficient condition for X_k:n ≥_st Y_k:n where st is the usual ordering for distributions. In the Markovian case (all components have an exponential lifetime), we give a necessary and sufficient condition. We prove that X_k:n is greater that Y_k:n according to the usual stochastic ordering if and only if