Optimal Confidence Regions for the Multinomial Parameter
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Construction of tight confidence regions and intervals is central to statistical inference and decision-making. Consider an empirical distribution p generated from n iid realizations of a random variable that takes one of k possible values according to an unknown distribution p. This is analogous with a single draw from a multinomial distribution. A confidence region is a subset of the probability simplex that depends on p and contains the unknown p with a specified confidence. This paper shows how one can construct minimum average volume confidence regions, answering a long standing question. We also show the optimality of the regions directly translates to optimal confidence intervals of functionals, such as the mean, variance and median.
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