Conditions for preserving transform order relations of identically distributed component lifetimes by system lifetimes

Abstract

We describe conditions for preserving relations with given distributions in the transform orders: convex transform, star, and superadditive ones of the marginal distributions of identically distributed component lifetimes by semicoherent system lifetimes. If the distinguished extremal element of the family has a left bounded support, then the conditions are necessary and sufficient. In particular, we specify the conditions for preserving the monotone generalized failure rates and their reversed versions, generalized monotone failure rates on the average, generalized new better and worse than used, and decreasing log-odds rate properties. The conditions depend on the minimal cut sets of the system and the copula of dependence among the component lifetimes. An alternative conditions are based on the respective minimal path sets and survival copula.