Hazard rate comparison of parallel systems with heterogeneous gamma components

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 λ 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 λ ̃ = ( ∏ i = 1 n λ i ) 1 / n , the geometric mean of λ 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.