Mean distances and dependence structures for lifetimes of systems with shared components

Distortion and copula functions represent powerful tools in the description of the reliability of some complex systems as functions of their components' reliability. On this aim we study several pairs of reliability systems with one or more shared components, in the case when their lifetimes are independent and identically distributed or independent but not identically distributed. We focus on the dependence that arises from sharing components, often described by Marshall-Olkin copulas, making use of some distance measures related to the Gini's mean difference and its new recent generalizations. A special role is played by a new distortion function related to the ROC curve.