Majorization and Stochastic Orders

Abstract

The simplest stochastic order is an ordering of ‘largeness’ on random variables or random vectors; more generally, stochastic orders are orderings that are used to compare random variables or random vectors, or probability distributions or measures. Majorization orderings are orderings of divergence from uniformity and provide a means of proving many inequalities. Several equivalent forms are given for the stochastic largeness order for random variables, and for measures in a partially ordered space, from which generalizations come from extending one or more of the equivalent forms. Similarly, several equivalent forms are given for vector majorization and generalized majorization for functions on a measure space. Dependence orderings that result from stochastic and majorization orderings are obtained and contrasted. Examples where the orderings overlap, including the Lorenz ordering, are mentioned. Applications to illustrate the orderings include measures of inequality and diversity, measures of dependence, preference orderings in random utility models for choice, and comparison of risks for portfolios.