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