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

• Volume: 10, page 1-10
• ISSN: 1292-8100

top

## Abstract

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

## How to cite

top

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 -

## References

top
1. [1] J.-L. Bon and E. Păltănea, Ordering properties of convolutions of exponential random variables. Lifetime Data Anal. 5 (1999) 185–192. Zbl0967.60017
2. [2] G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities. Cambridge University Press, Cambridge (1934). Zbl0010.10703JFM60.0169.01
3. [3] B.-E. Khaledi and S. Kochar, Some new results on stochastic comparisons of parallel systems. J. Appl. Probab. 37 (2000) 1123–1128. Zbl0995.62104
4. [4] A.W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications. Academic Press, New York (1979). Zbl0437.26007MR552278
5. [5] E. Păltănea. A note of stochastic comparison of fail-safe Markov systems, in $17$th Scientific Session on Mathematics and its Aplications, “Transilvania” Univ. Press (2003) 179–182.
6. [6] P. Pledger and F. Proschan, Comparisons of order statistics and spacing from heterogeneous distributions, in Optimizing Methods in Statistics. Academic Press, New York (1971) 89–113. Zbl0263.62062
7. [7] M. Shaked and J.G. Shanthikumar, Stochastic Orders and Their Applications. Academic Press, New York (1994). Zbl0806.62009MR1278322

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.