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 : 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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