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

A distribution function F is a generalized distorted distribution of the distribution functions $$F_1,\ldots ,F_n$$ if $$F=Q(F_1,\ldots ,F_n)$$ for an increasing continuous distortion function Q such that $$Q(0,\ldots ,0)=0$$ and $$Q(1,\ldots ,1)=1$$ . 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 $$F_1,\ldots ,F_n$$ 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. [ABSTRACT FROM AUTHOR]