Multivariate stochastic dominance with fixed dependence structure (Q1109659)

A stochastic dominance relation is a partial ordering of probability distributions given by \[ (*)\quad \int_{{\mathcal X}}\phi (x) dF(x)\quad \geq \quad \int_{{\mathcal X}}\phi (x) dG(x)\quad for\quad all\quad \phi \in \Phi \] where F and G denote probability measures and \(\Phi\) is a given set of real-valued functions. When \({\mathcal X}\) is the real line and \(\Phi\) is, e.g., the set of all increasing, all concave, or all completely monotone functions, simple conditions are well-known which are necessary and sufficient for (*). Further, when \({\mathcal X}={\mathbb{R}}^ d \)and F as well as G are distributions of d independent random variables (*) is equivalent to conditions on the univariate marginal distributions only. The present contribution extends one of the latter results: Let F and G be distributions of dependent variables which have the same dependence structure in terms of their copula, and let \(\Phi\) be the set of all componentwise increasing measurable functions \({\mathbb{R}}\) \(d\to {\mathbb{R}}\). Then, (*) if and only if the usual univariate dominance conditions hold for every marginal.