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

Ayodele Ighodaro, Thomas Santner and Lawrence Brown

Journal of Multivariate Analysis, 1982, vol. 12, issue 4, 469-479

Abstract: 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.

Keywords: Simultaneous; estimation; multinomial; distribution; minimal; complete; class; entropy; loss; square; error; loss; maximum; likelihood; estimator; admissible (search for similar items in EconPapers)
Date: 1982
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