Admissibility and complete class results for the multinomial estimation problem with entropy and squared error loss
Suppose \(X=(x_ 1,...,x_ t)\) has a multinomial distribution based on N trials with unknown vector of cell probabilities \(p=(p_ 1,...,p_ t)\) in \({\mathcal S}\equiv \{p: p_ i\geq 0\), \(i=1,...,t\), \(\sum p_ i=1\}\). This paper studies the problem of simultaneously estimating p under entropy loss (EL) and squared error loss (SEL). It is proved that \(\delta\) (X) is Bayes with respect to EL for prior P if and only if \(\delta\) (X) is Bayes w.r.t. SEL for P. The admissible rules under EL are shown to be Bayes. The minimal complete class of rules w.r.t. EL is characterized. As an application, the maximum likelihood estimator is proved to be admissible w.r.t. EL for the problem with parameter space \({\mathcal S}\) and inadmissible for the problem with parameter space \({\mathcal S}\) minus its vertices. It is shown that when \(N\geq t-1\), a positive estimator \(\delta\) is admissible under SEL if and only if \(\delta\) is admissible under EL. Examples are given of nonpositive estimators which are admissible under SEL but not under EL and vice versa.
scientific article; zbMATH DE number 3858154