Inference from Multinomial Data Based on a MLE-Dominance Criterion

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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].