Inference from Multinomial Data Based on a MLE-Dominance Criterion

Alessio Benavoli, Cassio Polpo de Campos

Abstract: We consider the problem of inference from multinomial data with chances $\boldsymbol{\theta}$ , subject to the a-priori information that the true parameter vector $\boldsymbol{\theta}$ belongs to a known convex polytope $\boldsymbol{\Theta}$ . The proposed estimator has the parametrized structure of the conditional-mean estimator with a prior Dirichlet distribution, whose parameters (s,t) are suitably designed via a dominance criterion so as to guarantee, for any $\boldsymbol{\theta} \in \boldsymbol{\Theta}$ , an improvement of the Mean Squared Error over the Maximum Likelihood Estimator (MLE). The solution of this MLE-dominance problem allows us to give a different interpretation of: (1) the several Bayesian estimators proposed in the literature for the problem of inference from multinomial data; (2) the Imprecise Dirichlet Model (IDM) developed by Walley [13].