Issue |
ESAIM: PS
Volume 10, September 2006
|
|
---|---|---|
Page(s) | 1 - 10 | |
DOI | https://doi.org/10.1051/ps:2005020 | |
Published online | 16 December 2005 |
Comparison of order statistics in a random sequence to the same statistics with i.i.d. variables
1
Polytech-Lille, USTL, Laboratoire CNRS Painlevé, 59655 Villeneuve d'Ascq, France; jean-louis.bon@polytech-lille.fr
2
Transilvania University of Braşov, Faculty of Mathematics and Computer Sciences, România.
Received:
14
December
2004
Revised:
25
May
2005
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 Ui,i = 1,...,n, be positive
independent random variables with common distribution F.
For λi > 0 and µ > 0, let consider
Xi = Ui/λi and Yi = Ui/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 Xk:n to be the kth
order statistics of the random variables X1,...,Xn, and
similarly Yk:n to be the kth order statistics of
Y1,...,Yn.
If Xi,i = 1,...,n, are the lifetimes of the components of a
n+1-k-out-of-n non-repairable system, then Xk:n is the
lifetime of the system.
In this paper, we give for a fixed k a sufficient condition for
Xk:n ≥st Yk: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 Xk:n is greater that Yk:n according to the usual
stochastic ordering if and only if
Mathematics Subject Classification: 60E15 / 62N05 / 62G30 / 90B25 / 60J27
Key words: Stochastic ordering / Markov system / order statistics / k-out-of-n.
© EDP Sciences, SMAI, 2006
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