Likelihood ratio ordering of convolutions of heterogeneous exponential and geometric random variables

In this paper, we study the convolutions of heterogeneous exponential and geometric random variables in terms of the weakly majorization order () of parameter vectors and the likelihood ratio order (>=lr). It is proved that order between two parameter vectors implies >=lr order between convolutions of two heterogeneous exponential (geometric) samples. For the two-dimensional case, it is found that there exist stronger equivalent characterizations. These results strengthen the corresponding ones of Boland et al. [Boland, P.J., El-Neweihi, E., Proschan, F., 1994. Schur properties of convolutions of exponential and geometric random variables. Journal of Multivariate Analysis 48, 157-167] by relaxing the conditions on parameter vectors from the majorization order () to order.