Stochastic comparisons of distorted distributions, coherent systems and mixtures with ordered components

摘要

A distribution function F is a generalized distorted distribution of the distribution functions F-1, ..., F-n if F = Q(F-1, ..., F-n) for an increasing continuous distortion function Q such that and Q(1, ..., 1) = 1. In this paper, necessary and sufficient conditions for the stochastic (ST) and the hazard rate (HR) orderings of generalized distorted distributions are provided when the distributions F-1, ..., F-n are ordered. These results are used to obtain distribution-free ordering properties for coherent systems with heterogeneous components. In particular, we determine all the ST and HR orderings for coherent systems with 1-3 independent components. We also compare systems with dependent components. The results on distorted distributions are also used to get comparisons of finite mixtures.