ESAIM: Probability and Statistics (ESAIM: P&S)

ESAIM: P&S, January 2006, Vol. 10, pp. 1-10
DOI: 10.1051/ps:2005020

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

Jean-Louis Bon¹ and Eugen Paltanea²

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

(Received December 14, 2004. Revised May 25, 2005. / Published online: 16 December 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 $\lambda_i>0$ and $\mu >0$ , let consider $X_i=U_i/\lambda_i$ and $Y_i=U_i/\mu,\ i=1,...,n$ . Remark that this is no more than a change of scale for each term. For $k\in\{1,2,...,n\},$ 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 $X_{k:n}\geq_{st}Y_{k:n}$ 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

$\begin{displaymath}\left( \begin{array}{c} n k \end{array}\right) {\mu}^k \geq \... ...1<i_2<...<i_k\leq n}\lambda_{i_1}\lambda_{i_2}...\lambda_{i_k}.\end{displaymath}$

Mathematics Subject Classification. 60E15, 62N05, 62G30, 90B25, 60J27.

Key words: Stochastic ordering, Markov system, order statistics, k-out-of-n.

© EDP Sciences, SMAI 2006