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

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 Ui,i = 1,...,n, be positive independent random variables with common distribution F. For λi > 0 and µ > 0, let consider Xi = Ui/λi and Yi = Ui/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 Xk:n to be the kth order statistics of the random variables X1,...,Xn, and similarly Yk:n to be the kth order statistics of Y1,...,Yn. If Xi,i = 1,...,n, are the lifetimes of the components of a n+1-k-out-of-n non-repairable system, then Xk:n is the lifetime of the system. In this paper, we give for a fixed k a sufficient condition for Xk:n ≥st Yk: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 Xk:n is greater that Yk: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<...