Inequalities for Distributions with Given Marginals

An ordering on discrete bivariate distributions formalizing the notion of concordance is defined and shown to be equivalent to stochastic ordering of distribution functions with identical marginals. Furthermore, for this ordering, ∫φ dH is shown to be H-monotone for all superadditive functions φ, generalizing earlier results of Hoeffding, Frechet, Lehmann and others. The usual correlation coefficient, Kendall's τ and Spearman's ρ are shown to be monotone functions of H. That ∫φ dH is H-monotone holds for distributions on Rⁿ with fixed $(n - 1)$-dimensional marginals for any φ with nonnegative finite differences of order n. Some related results are obtained. Stochastic ordering is preserved under certain transformations, e.g., convolutions. A distribution on R^∞ is constructed, making max(X₁,⋯, Xₙ) stochastically largest for all n when Xᵢ have given one-dimensional distributions, generalizing a result of Robbins. Finally an ordering for doubly stochastic matrices is proposed.