A note on the limiting behaviour of hazard rate functions of generalized mixtures

Abstract

The mixtures are popular models in probability and statistics. They represent populations made up with different groups. Their distributions are linear combinations with positive weights of the baseline distribution functions. They can be extended by allowing that some of these weights could be negative. The linear combinations with some negative weights are called “negative mixtures” and they appear in different contexts. For example, the distribution of a coherent system can be represented as a negative mixture of the distributions of series (or parallel) systems made up with their components. The same happen with the finite sums of random variables both when they are independent (convolution) or dependent (C-convolution). In general, it is not easy to determine the shape of the reliability and hazard rate functions of mixtures. The same can be applied to their mean residual life functions. However, some results have been obtained for the limiting behaviour when the time goes on (i.e. t→∞). Here we provide new results in this direction that can be applied both to the classical case of mixtures with positive weights and the other cases that contains some negative weights. Some illustrative examples show that these new results allow us to solve cases that cannot be solved with the preceding findings.