ON PARALLEL SYSTEMS WITH HETEROGENEOUS GAMMA COMPONENTS

摘要

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 X_1 and X_2 be two independent gamma random variables with X_i having shape parameter r > 0 and scale parameter λ_i, 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 X_i~*,i = 1,2. Denote by X_(2:2) and X_(2:2)* the corresponding maximum order statistics, respectively. It is proved that, among others, if (λ_1,λ_2) weakly majorize (X_1~*, X_2~*), then X_(2:2) 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 (λ_1,λ_2) is p-larger than (λ_1~*,λ_2~*), then X_(2:2) 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.