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

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