🤖 AI Summary
This study addresses the challenge of effectively disentangling and quantifying the individual contributions of marginal effects and dependence structure in multivariate statistical functionals under arbitrary marginal distributions, while establishing sharp extremal bounds when the dependence structure is uncertain. To this end, the authors propose a conditional copula–based decomposition that unifies the representation of multivariate functional expectations as combinations of marginal distributions and conditional copulas, thereby constructing a quantile–copula analytical framework. By introducing Δ-antitonicity to characterize the class of functionals amenable to sharp bounds, and integrating concordance order with functional integration techniques, the work extends the applicability of copula theory in uncertainty modeling. The framework’s generality and the tightness of its extremal bounds are demonstrated through applications in risk measures, stochastic dominance probabilities, information entropy, and option pricing.
📝 Abstract
In this paper, we derive a conditional copula representation for expectations of the form $\mathbb{E}[g(\boldsymbol{X})]$, where $\boldsymbol{X}$ is a random vector with arbitrary marginal distributions and $g$ is a measurable function satisfying suitable integrability conditions. The proposed representation explicitly separates the contributions of the marginal distributions and the dependence structure through conditional copula distributions, yielding a unified quantile--copula framework for a broad class of statistical functionals. This framework encompasses numerous quantities of practical interest, including moments, probabilities, dependence measures, inequality indices, entropy measures, and multivariate functionals. We further establish extremal bounds under fixed marginals by exploiting the concordance order on copulas and characterize the classes of functions for which these bounds apply through the notion of $Δ$-antitonicity. Finally, several illustrative examples illustrate the versatility of the proposed framework through applications to risk measures, stochastic superiority probabilities, information measures, and option pricing under dependence uncertainty.