On Parallel Systems With Heterogeneous Gamma Components

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 λ * 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 (λ * 1 , λ * 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.