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 X1,…,Xn be independent gamma random variables with Xi having shape parameter 0<r≤1 and scale parameter λi, i=1,…,n. Denote by Yn:n the largest order statistic arising from i.i.d. gamma random variables Y1,…,Yn with Yi having shape parameter r and scale parameter λ̃=(∏i=1nλi)1/n, the geometric mean of λi′s. It is shown that Xn:n is stochastically larger than Yn: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.