AI 中文总结
本文推导了多元统计泛函期望的条件Copula表示，建立了固定边缘下的极值界，通过示例展示了其在风险测度等多领域应用的通用性。
AI 中文摘要
本文推导了形如$\boldsymbol{E}[g(\boldsymbol{X})]$的期望的条件Copula表示，其中$\boldsymbol{X}$是具有任意边缘分布的随机向量，$g$是满足适当可积条件的可测函数。所提出的表示通过条件Copula分布明确区分了边缘分布和相依结构的贡献，为广泛类别的统计泛函提供了统一的分位数-Copula框架。该框架涵盖了众多具有实际意义的量，包括矩、概率、相依测度、不平等指数、熵测度和多元泛函。我们进一步利用Copula的一致性序建立了固定边缘下的极值界，并通过$\boldsymbol{\triangle}$-反序性的概念刻画了这些界适用的函数类。最后，多个示例展示了所提框架在风险测度、随机优势概率、信息测度及相依不确定性下的期权定价等应用中的通用性。
英文摘要
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.
Comments11 pages
