Abstract: | Let X ≡ (X1, …, Xt) have a multinomial distribution based on N trials with unknown vector of cell probabilities p ≡ (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 δ is Bayes w.r.t. EL for prior P if and only if δ 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 = j=1kFj where k ≥ 1 and each Fj is a facet of
which satisfies: F a facet of
such that F naFjF 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, ξ(x) ≡ ∫ f(x¦p) P(dp) > 0 for all x in {x
: sup
0 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 Θ =
but it is shown to be inadmissible for the problem with parameter space Θ = (
minus its vertices). This is a severe form of “tyranny of boundary.” Finally it is shown that when N ≥ t − 1 any estimator δ which satisfies δ(x) > 0 x
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. |