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 ,i=1,...,n, be positive independent random variables with common distribution F. For λ i >0 and μ>0, let consider X i =U i /λ i and Y i =U i /μ,i=1,...,n. Remark that this is no more than a change of scale for each term. For k{1,2,...,n}, let us define X k:n to be the kth order statistics of the random variables X 1 ,...,X n , and similarly Y k:n to be the kth 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 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

nkμ k 1i 1 <i 2 <...<i k n λ i 1 λ i 2 ...λ i k .

DOI : 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},
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     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/

[1] J.-L. Bon and E. Păltănea, Ordering properties of convolutions of exponential random variables. Lifetime Data Anal. 5 (1999) 185-192. | Zbl

[2] G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities. Cambridge University Press, Cambridge (1934). | JFM | Zbl

[3] B.-E. Khaledi and S. Kochar, Some new results on stochastic comparisons of parallel systems. J. Appl. Probab. 37 (2000) 1123-1128. | Zbl

[4] A.W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications. Academic Press, New York (1979). | MR | Zbl

[5] E. Păltănea. A note of stochastic comparison of fail-safe Markov systems2003) 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. | Zbl

[7] M. Shaked and J.G. Shanthikumar, Stochastic Orders and Their Applications. Academic Press, New York (1994). | MR | Zbl

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