| 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 X 1,...,X n, and similarly Y k:n to be the kth order statistics of Y 1,...,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 \[\left( \begin{array}{c} n k end{array}\right) {\mu}^k \geq \sum_{1\leq i_1<i_2<...<i_k\leq n}\lambda_{i_1}\lambda_{i_2}...\lambda_{i_k}.\].
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