Schur convexity of the maximum likelihood function for the multivariate hypergeometric and multinomial distributions

Abstract

We define for a family distributions p θ ( x), θ ϵ Θ, the maximum likelihood function L at a sample point x by L( x) = sup θϵΘPθ ( x). We show that for the multivariate hypergeometric and multinomial families, the maximum likelihood function is a Schur convex function of x. In the language of majorization, this implies that the more diverse the elements or components of x are, the larger is the function L( x). Several applications of this result are given in the areas of parameter estimation and combinatorics. An improvement and generalization of a classical inequality of Khintchine is also derived as a consequence.