Stochastic orders for convolution of heterogeneous gamma and negative binomial random variables

Convolutions of independent random variables often arise in a natural way in many applied problems. In this article, we compare convolutions of two sets of gamma (negative binomial) random variables in the convolution order and the usual stochastic order in a unified set-up, when the shape and scale parameters satisfy a partial order called reverse-coupled majorization order. This partial order is an extension of the majorization order from vectors to pairs of vectors which also incorporates the arrangement information. The results established in this article strengthen those known in the literature.