Comparison of order statistics in a random sequence to the same statistics with I.I.D. variables
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 ${U}_{i},\phantom{\rule{0.166667em}{0ex}}i=1,...,n,$ 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 ${Y}_{i}={U}_{i}/\mu ,\phantom{\rule{4pt}{0ex}}i=1,...,n$. Remark that this is no more than a change of scale for each term. For $k\in \left\{1,2,...,n\right\},$ 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 ${X}_{i},\phantom{\rule{4pt}{0ex}}i=1,...,n,$ 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

 $\left(\begin{array}{c}n\\ k\end{array}\right){\mu }^{k}\ge \sum _{1\le {i}_{1}<{i}_{2}<...<{i}_{k}\le n}{\lambda }_{{i}_{1}}{\lambda }_{{i}_{2}}...{\lambda }_{{i}_{k}}.$

DOI : https://doi.org/10.1051/ps:2005020
Classification : 60E15,  62N05,  62G30,  90B25,  60J27
Mots clés : stochastic ordering, Markov system, order statistics, $k$-out-of-$n$
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author = {Bon, Jean-Louis and P\u{a}lt\u{a}nea, Eugen},
title = {Comparison of order statistics in a random sequence to the same statistics with {I.I.D.} variables},
journal = {ESAIM: Probability and Statistics},
pages = {1--10},
publisher = {EDP-Sciences},
volume = {10},
year = {2006},
doi = {10.1051/ps:2005020},
mrnumber = {2188345},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps:2005020/}
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Bon, Jean-Louis; Păltănea, Eugen. Comparison of order statistics in a random sequence to the same statistics with I.I.D. variables. ESAIM: Probability and Statistics, Tome 10 (2006), pp. 1-10. doi : 10.1051/ps:2005020. http://www.numdam.org/articles/10.1051/ps:2005020/

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