Ordering properties of convolutions of exponential random variables

Convolutions of independent random variables are usually compared. In this paper, after a synthetic comparison with respect to hazard rate ordering between sums of independent exponential random variables, we focus on the special case where one sum is identically distributed. So, for a given sum of n independent exponential random variables, we deduce the "best" Erlang-n bounds, with respect to each of the usual orderings: mean ordering, stochastic ordering, hazard rate ordering and likelihood ratio ordering.