Second-order stochastic comparisons of order statistics

We study the problem of comparing ageing patterns lifetimes k-out-of-n systems with i.i.d. components. Mathematically, this reduces to being able decide about a stochastic ordering relationship between different order statistics. discuss such relationships respect second-order dominance, obtaining characterizations through verification relative convexity suitably chosen reference distribution function. introduce hierarchy functions leading classes, each expressing and increasing knowledge precision components lifetimes. Such classes are wide enough include popular families distributions, as, for example, failure rate distributions. derive sufficient dominance conditions depending on identification class which includes component applicability method characterize test convexity, as notion plays central role in proposed approach.