ORDERING CONVOLUTIONS OF HETEROGENEOUS EXPONENTIAL AND GEOMETRIC DISTRIBUTIONS REVISITED

Let Sn(a1, …, an) be the sum of n independent exponential random variables with respective hazard rates a1, …, an or the sum of n independent geometric random variables with respective parameters a1, …, an. In this article, we investigate sufficient conditions on parameter vectors (a1, …, an) and $(a_{1}^{\ast},\ldots,a_{n}^{\ast})$ under which Sn(a1, …, an) and $S_{n}(a_{1}^{\ast},\ldots,a_{n}^{\ast})$ are ordered in terms of the increasing convex and the reversed hazard rate orders for both exponential and geometric random variables and in terms of the mean residual life order for geometric variables. For the bivariate case, all of these sufficient conditions are also necessary. These characterizations are used to compare fail-safe systems with heterogeneous exponential components in the sense of the increasing convex and the reversed hazard rate orders. The main results complement several known ones in the literature.