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 $\Delta$-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.
Let $Y_1,\ldots,Y_n$ be independent symmetric random variables with log-concave tails. We give a dimension-free characterization of the expected supremum of the canonical process $X_x=\sum_{i=1}^n x_iY_i$ without any $\Delta_2$ or regular-growth assumption on the coordinate tails. The characterization is governed by sc...
Entropy functionals and their associated divergences underlie many statistical methods, including maximum entropy inference, minimum divergence estimation, and goodness-of-fit testing, yet choosing among Shannon, R\'enyi, Tsallis, and more general entropies is often a matter of convention rather than structural princip...
Testing whether two independent samples arise from the same underlying distribution is a fundamental statistical problem. We propose a new class of two-sample distribution tests based on a family of probability metrics $\Delta_{n,p}$, constructed from repeatedly integrated quantile functions. On their respective domain...
Zhen-Feng Zou, Meng Guan, Pan-Xu Yuan et al.· 0 citations
This work introduces the class $\mathcal{P}_\psi(\mu)$ of measures that differ from a reference measure only through a finite-dimensional map $\psi$ while preserving the reference conditionals on its fibers, and develops approximation theory for fitting within it.
R. Baptista, Bamdad Hosseini, Alexander Hsu· 0 citations
Let $\Pi$ denote the independence copula and $M,W$ the upper and lower Fr\'echet--Hoeffding copulas. Catalano and Lavenant (2025) conjectured that $M$ and $W$ maximize the quadratic Wasserstein distance from $\Pi$ among all bivariate copulas. We prove this conjecture and characterize all equality cases: $\mathcal W_2^2...
We show that for positive semidefinite $A$ and $B$ the relative entropy can be written as $D(A\|B)=2\int a\log(a/b)\,\mathrm{d}\mu_{A,B}(a,b)$, where $\mu_{A,B}$ denotes Hein\"avaara's tracial joint spectral measure. This allows us to obtain a unified derivation of several known integral representations of relative ent...
Lukas Schmitt· 0 citations
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