Comparison of order statistics in a random sequence to the same statistics with I.I.D. variables

Bon, Jean-Louis; Păltănea, Eugen

ESAIM: Probability and Statistics, Tome 10 (2006), pp. 1-10.

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 $k$th order statistics of the random variables $X_1,...,X_n$, and similarly $Y_{k:n}$ to be the $k$th 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 $X_{k:n}{\ge }_{st}{Y}_{k:n}$ 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

$$\left(\begin{array}{c}n\\ k\end{array}\right){\mu }^k\ge \sum _{1\le i_1<i_2<...<i_k\le n}{\lambda }_{i_1}{\lambda }_{i_2}...{\lambda }_{i_k}.$$

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