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

Bon, Jean-Louis, and Păltănea, Eugen. "Comparison of order statistics in a random sequence to the same statistics with I.I.D. variables." ESAIM: Probability and Statistics 10 (2006): 1-10. <http://eudml.org/doc/245652>.

@article{Bon2006,
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 $\lambda _i&gt;0$ and $\mu &gt;0$, 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 $k\in \lbrace 1,2,...,n\rbrace ,$ 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\[\left( \begin\{array\}\{c\} n\\k \end\{array\}\right) \{\mu \}^k \ge \sum \_\{1\le i\_1&lt;i\_2&lt;...&lt;i\_k\le n\}\lambda \_\{i\_1\}\lambda \_\{i\_2\}...\lambda \_\{i\_k\}.\]},
author = {Bon, Jean-Louis, Păltănea, Eugen},
journal = {ESAIM: Probability and Statistics},
keywords = {stochastic ordering; Markov system; order statistics; $k$-out-of-$n$; Stochastic ordering; -out-of-},
language = {eng},
pages = {1-10},
publisher = {EDP-Sciences},
title = {Comparison of order statistics in a random sequence to the same statistics with I.I.D. variables},
url = {http://eudml.org/doc/245652},
volume = {10},
year = {2006},
}

TY - JOUR
AU - Bon, Jean-Louis
AU - Păltănea, Eugen
TI - Comparison of order statistics in a random sequence to the same statistics with I.I.D. variables
JO - ESAIM: Probability and Statistics
PY - 2006
PB - EDP-Sciences
VL - 10
SP - 1
EP - 10
AB - 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 $\lambda _i&gt;0$ and $\mu &gt;0$, 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 $k\in \lbrace 1,2,...,n\rbrace ,$ 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\[\left( \begin{array}{c} n\\k \end{array}\right) {\mu }^k \ge \sum _{1\le i_1&lt;i_2&lt;...&lt;i_k\le n}\lambda _{i_1}\lambda _{i_2}...\lambda _{i_k}.\]
LA - eng
KW - stochastic ordering; Markov system; order statistics; $k$-out-of-$n$; Stochastic ordering; -out-of-
UR - http://eudml.org/doc/245652
ER -