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

Jean-Louis Bon; Eugen Păltănea

ESAIM: Probability and Statistics (2006)

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

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topBon, 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 $\lambda _i>0$ 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 $k\in \lbrace 1,2,...,n\rbrace ,$ 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\[\left( \begin\{array\}\{c\} n\\k \end\{array\}\right) \{\mu \}^k \ge \sum \_\{1\le i\_1<i\_2<...<i\_k\le n\}\lambda \_\{i\_1\}\lambda \_\{i\_2\}...\lambda \_\{i\_k\}.\]},

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 $\lambda _i>0$ 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 $k\in \lbrace 1,2,...,n\rbrace ,$ 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\[\left( \begin{array}{c} n\\k \end{array}\right) {\mu }^k \ge \sum _{1\le i_1<i_2<...<i_k\le n}\lambda _{i_1}\lambda _{i_2}...\lambda _{i_k}.\]

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] J.-L. Bon and E. Păltănea, Ordering properties of convolutions of exponential random variables. Lifetime Data Anal. 5 (1999) 185–192. Zbl0967.60017
- [2] G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities. Cambridge University Press, Cambridge (1934). Zbl0010.10703JFM60.0169.01
- [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] A.W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications. Academic Press, New York (1979). Zbl0437.26007MR552278
- [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] 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] M. Shaked and J.G. Shanthikumar, Stochastic Orders and Their Applications. Academic Press, New York (1994). Zbl0806.62009MR1278322

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