Stochastic Ordering of Dependent Systems under Transformation Models and Archimedean Copulas

TL;DR

This paper investigates how dependence and heterogeneity in system components, modeled via Archimedean copulas and transformations, influence stochastic ordering of system lifetimes, providing new criteria for reliability comparison.

Contribution

It introduces unified conditions for stochastic dominance in dependent systems using transformation models and copula generator properties, extending beyond independence assumptions.

Findings

Derived conditions for stochastic dominance in dependent systems.

Extended reliability comparisons to models with dependence and heterogeneity.

Provided tractable criteria based on copula generator properties.

Abstract

We study stochastic ordering of system lifetimes with dependent and heterogeneous components whose marginal distributions are obtained through transformations of a common baseline. The dependence structure is modeled via Archimedean copulas, allowing for a unified treatment of several transformation-based models, including proportional hazard, proportional reversed hazard rate and proportional odds families. For parallel, series and $(n-k)$-out-of-$n$ systems, we derive conditions for stochastic dominance based on monotonicity of the transformation and structural properties of the copula generators, formulated through super-additivity and Schur-type arguments. The results provide tractable criteria that extend existing comparisons beyond independence and illustrate the combined effect of dependence and parameter heterogeneity on system reliability.