Admissibility and complete class results for the multinomial estimation problem with entropy and squared error loss
type: Item
instance of: scholarly article
title: Admissibility and complete class results for the multinomial estimation problem with entropy and squared error loss
zbMATH Open document ID: 0539.62009
DOI: 10.1016/0047-259X(82)90057-4
author: Ayodele Ighodaro
author: Thomas Santner
author: Lawrence Brown
published in: Journal of Multivariate Analysis
publication date: 1982-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 62C15
Mathematics Subject Classification ID: 62C07
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 62F10
zbMATH DE Number: 3858154
zbMATH Keywords: Bayes rule
zbMATH Keywords: multinomial distribution
zbMATH Keywords: entropy loss
zbMATH Keywords: squared error loss
zbMATH Keywords: minimal complete class
zbMATH Keywords: maximum likelihood estimator
zbMATH Keywords: nonpositive estimators
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3858154