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Two sufficient conditions for comparing convolutions of heterogeneous gamma random variables in terms of star order are established. It is further shown that if the scale parameters of heterogeneous gamma random variables are more dispersed in the sense of majorization, then the convolutions are more dispersed according to the right spread order, which generalizes and strengthens the results in Diaconis and Perlman (1987), Kochar and Xu (2010) and Zhao and Balakrishnan (2009).

This note investigates the negative dependence in frailty models. First, we show that the frailty variable and the overall population variable are negatively likelihood ratio dependent and derive an upper bound for the survival function of the population with higher frailty. Secondly, we prove that the DFR property and the logconvex hazard rate of the baseline variable imply the DLR property of the population variable. Finally, we further prove that the likelihood ratio order, hazard rate order and reversed hazard rate order between two frailty variables imply the likelihood ratio order, reversed hazard rate order, and hazard rate order between the corresponding overall population variables, respectively.

For two non-negative random variables X and Y, it is shown that if X is smaller than Y in the sense of the likelihood ratio order and either X or Y is of decreasing likelihood ratio, then the m-spacing V k : n ( m ) of an X-sample is smaller than W k : n ( m ) , the m-spacing of a Y-sample, in terms of the likelihood ratio order. It is also proved that a similar result for the upshifted likelihood ratio order is valid. Finally, it is proved that the decreasing failure rate in average property of a population X is preserved by sample spacing V k : n ( 1 ) .