# 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

## Access Full Article

top## Abstract

top## How to cite

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 (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

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

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.