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ON PARALLEL SYSTEMS WITH HETEROGENEOUS GAMMA COMPONENTS

Published online by Cambridge University Press:  17 May 2011

Peng Zhao
Peng Zhao
Affiliation:
School of Mathematics and Statistics Lanzhou University, Lanzhou 730000, China E-mail: [email protected]
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Abstract

In this article, we study ordering properties of lifetimes of parallel systems with two independent heterogeneous gamma components in terms of the likelihood ratio order and the hazard rate order. Let X1 and X2 be two independent gamma random variables with Xi having shape parameter r>0 and scale parameter λi, i=1, 2, and let X*1 and X*2 be another set of independent gamma random variables with X*i having shape parameter r and scale parameter λ*i, i=1, 2. Denote by X2:2 and X*2:2 the corresponding maximum order statistics, respectively. It is proved that, among others, if (λ1, λ2) weakly majorize (λ*1, λ*2), then X2:2 is stochastically greater than X*2:2 in the sense of likelihood ratio order. We also establish, among others, that if 0<r≤1 and (λ1, λ2) is p-larger than (λ*1, λ*2), then X2:2 is stochastically greater than X*2:2 in the sense of hazard rate order. The results derived here strengthen and generalize some of the results known in the literature.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 25 , Issue 3 , July 2011 , pp. 369 - 391
Copyright
Copyright © Cambridge University Press 2011

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