Majorization, Randomness and Dependence for Multivariate Distributions

Abstract

The preorder relation of Hardy, Littlewood and Polya (1929), Day (1973) and Chong (1974, 1976) is applied to multivariate probability densities. This preorder, which is called majorization here, can be interpreted as an ordering of randomness. When used to compare multivariate densities with the same marginal densities, it can be interpreted as an ordering of dependence or conditional dependence. Results in Hickey (1983, 1984) and Joe (1985) are generalized. A relative entropy function is proposed as a measure of dependence or conditional dependence for multivariate densities with the same marginals.