Hazard rate comparison of parallel systems with heterogeneous gamma components

JOURNAL OF MULTIVARIATE ANALYSIS

卷:113

Article

Balakrishnan, N.¹  Zhao, Peng² 

[1] McMaster Univ, Dept Math & Stat, Hamilton, ON L8S 4K1, Canada

[2] Lanzhou Univ, Sch Math & Stat, Lanzhou 730000, Peoples R China

关键词: Gamma distribution;    Stochastic order;    Hazard rate order;    Order statistics;    Parallel system;   

DOI  :  10.1016/j.jmva.2011.05.001

来源: Elsevier

【 摘 要 】

We compare the hazard rate functions of the largest order statistic arising from independent heterogeneous gamma random variables and that arising from i.i.d. gamma random variables. Specifically, let X-1...., X-n be independent gamma random variables with X-i having shape parameter 0 < r <= 1 and scale parameter lambda(i), i = 1,..., n. Denote by Y-n:n the largest order statistic arising from i.i.d. gamma random variables Y-1,..., Y-n with Y-i having shape parameter r and scale parameter <(lambda)over bar> = (Pi(n)(i=1) lambda(i))(1/n), the geometric mean of lambda(i)'s. It is shown that X-n:n is stochastically larger than Y-n:n in terms of hazard rate order. The result derived here strengthens and generalizes some of the results known in the literature and leads to a sharp upper bound on the hazard rate function of the largest order statistic from heterogeneous gamma variables in terms of that of the largest order statistic from i.i.d. gamma variables. A numerical example is Finally provided to illustrate the main result established here. (C) 2011 Elsevier Inc. All rights reserved.

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