Stochastic comparisons of generalized mixtures and coherent systems

A distribution function F is a generalized mixture of the distribution functions F-1, ..., F-k if F = w(1)F(1)+ ... + w(k) F-k, where w(1),..., w(k) are some real numbers (weights) which should satisfy w(1) + ... + w(k) = 1. If all the weights are positive, then we have a classical finite mixture. If some weights are negative, then we have a negative mixture. Negative mixtures appear in different applied probability models (order statistics, estimators, coherent systems, etc.). The conditions to obtain stochastic comparisons of classical (positive) mixtures are well known in the literature. However, for negative mixtures, there are only results for the usual stochastic order. In this paper, conditions for hazard rate and likelihood ratio comparisons of generalized mixtures are obtained. These theoretical results are applied in this paper to study distribution-free comparisons of coherent systems using their representations as generalized mixtures. They can also be applied to other probability models in which the generalized mixtures appear.