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 (2005)

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

Abstract

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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 n k μ k ≥ ∑ 1 ≤ i 1 < i 2 < . . . < i k ≤ n λ i 1 λ i 2 . . . λ i k .

How to cite

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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 (2005): 1-10. <http://eudml.org/doc/104348>.

@article{Bon2005,
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 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 \[\left( \begin\{array\}\{c\} n k \end\{array\}\right) \{\mu\}^k \geq \sum\_\{1\leq i\_1<i\_2<...<i\_k\leq n\}\lambda\_\{i\_1\}\lambda\_\{i\_2\}...\lambda\_\{i\_k\}.\]},
author = {Bon, Jean-Louis, Păltănea, Eugen},
journal = {ESAIM: Probability and Statistics},
keywords = {Stochastic ordering; Markov system; order statistics; k-out-of-n.; -out-of-},
language = {eng},
month = {12},
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/104348},
volume = {10},
year = {2005},
}

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
DA - 2005/12//
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 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 \[\left( \begin{array}{c} n k \end{array}\right) {\mu}^k \geq \sum_{1\leq i_1<i_2<...<i_k\leq n}\lambda_{i_1}\lambda_{i_2}...\lambda_{i_k}.\]
LA - eng
KW - Stochastic ordering; Markov system; order statistics; k-out-of-n.; -out-of-
UR - http://eudml.org/doc/104348
ER -

References

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  1. J.-L. Bon and E. Păltănea, Ordering properties of convolutions of exponential random variables. Lifetime Data Anal. 5 (1999) 185–192.  
  2. G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities. Cambridge University Press, Cambridge (1934).  
  3. B.-E. Khaledi and S. Kochar, Some new results on stochastic comparisons of parallel systems. J. Appl. Probab.37 (2000) 1123–1128.  
  4. A.W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications. Academic Press, New York (1979).  
  5. E. Păltănea. A note of stochastic comparison of fail-safe Markov systems, in 17th 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.  
  7. M. Shaked and J.G. Shanthikumar, Stochastic Orders and Their Applications. Academic Press, New York (1994).  

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