Compound Multinomial Likelihood Functions are Unimodal: Proof of a Conjecture of I. J. Good

Abstract

I. J. Good's 1965 conjecture of the unimodality of the likelihood function of a symmetrical compound multinomial distribution is proved by the variation-diminishing property of the Laplace transform. The result is a special case of a several sample version with asymmetrical compounding Dirichlet distributions. The technique of proof is applied to yield similar results for the negative binomial distribution and a two point mixture of Poissons.