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 etÂ 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.
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