On hazard rate ordering of the sums of heterogeneous geometric random variables

In this paper, we treat convolutions of heterogeneous geometric random variables with respect to the p-larger order and the hazard rate order. It is shown that the p-larger order between two parameter vectors implies the hazard rate order between convolutions of two heterogeneous geometric sequences. Specially in the two-dimensional case, we present an equivalent characterization. The case when one convolution involves identically distributed variables is discussed, and we reveal the link between the hazard rate order of convolutions and the geometric mean of parameters. Finally, we drive the "best negative binomial bounds" for the hazard rate function of any convolution of geometric sequence under this setup.