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

We define for a family distributions p[theta](x), [theta] [epsilon] [Theta], the maximum likelihood function L at a sample point x by L(x) = sup[theta][epsilon][Theta]P[theta](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.