Admissibility and complete class results for the multinomial estimation problem with entropy and squared error loss

Let X [reverse not equivalent] (X1, ..., Xt) have a multinomial distribution based on N trials with unknown vector of cell probabilities p [reverse not equivalent] (p1, ..., pt). This paper derives admissibility and complete class results for the problem of simultaneously estimating p under entropy loss (EL) and squared error loss (SEL). Let and f(x|p) denote the (t - 1)-dimensional simplex, the support of X and the probability mass function of X, respectively. First it is shown that [delta] is Bayes w.r.t. EL for prior P if and only if [delta] is Bayes w.r.t. SEL for P. The admissible rules under EL are proved to be Bayes, a result known for the case of SEL. Let Q denote the class of subsets of of the form T = [up curve]j=1kFj where k >= 1 and each Fj is a facet of which satisfies: F a facet of such that F naFj=>F ncT. The minimal complete class of rules w.r.t. EL when N >= t - 1 is characterized as the class of Bayes rules with respect to priors P which satisfy P(0) = 1, [xi](x) [reverse not equivalent] [integral operator] f(x|p) P(dp) > 0 for all x in {x[set membership, variant]: sup0 f(x|p) > 0} for some 0 in Q containing all the vertices of . As an application, the maximum likelihood estimator is proved to be admissible w.r.t. EL when the estimation problem has parameter space [Theta] = but it is shown to be inadmissible for the problem with parameter space [Theta] = ( minus its vertices). This is a severe form of "tyranny of boundary." Finally it is shown that when N >= t - 1 any estimator [delta] which satisfies [delta](x) > 0 [for all]x [set membership, variant] is admissible under EL if and only if it is admissible under SEL. Examples are given of nonpositive estimators which are admissible under SEL but not under EL and vice versa.