Simultaneous estimation of Poisson means under entropy loss
type: Item
instance of: scholarly article
title: Simultaneous estimation of Poisson means under entropy loss
zbMATH Open document ID: 0654.62012
author: Malay Ghosh
author: Ming-Chung Yang
published in: The Annals of Statistics
publication date: 1988-01-01T00:00:00Z
review text: Let \(x=(x_ 1,...,x_ p)\) be p independent Poisson random variables with means \(\theta =(\theta_ 1,...,\theta_ p)\). For the problem of estimating \(\theta\) when the loss is the entropy distance given by \[ L(\theta,a)=\sum^{p}_{1}\theta_ i(a_ i/\theta_ i-\log (a_ i/\theta_ i)-1) \] where \(a=(a_ 1,...,a_ p)\), consider a class of estimators of the form \(XC+b\), where \(C=diag(c_ 1,...,c_ K)\) and \(b=(b_ 1,...,b_ K)\) are constants. It is shown that \(XC+b\) is an admissible estimator of \(\theta\) if and only if (i) \(b_ i>0\) and \(0\leq c_ i\leq 1\) for \(i=1,...,K\) and (ii) \(\sum_{c_ i=1}b_ i\leq 1\). It follows that \(X+\underset \tilde{} 1\) is inadmissible for \(K\geq 2\), where \(\underset \tilde{} 1\) denotes the vector (1,...,1). Monte Carlo results are given showing the risk improvement of certain estimators which dominate \(X+\underset \tilde{} 1\). It is noted that estimators of the form \(X+b\) are generalized Bayes with respect to improper (limiting gamma) priors.
Mathematics Subject Classification ID: 62C15
Mathematics Subject Classification ID: 62F10
Mathematics Subject Classification ID: 62C10
zbMATH DE Number: 4068053
zbMATH Keywords: simultaneous estimation of Poisson means
zbMATH Keywords: hierarchical Bayes estimators
zbMATH Keywords: entropy loss
zbMATH Keywords: Kullback-Leibler information
zbMATH Keywords: linear estimators
zbMATH Keywords: empirical Bayes
zbMATH Keywords: independent Poisson random variables
zbMATH Keywords: entropy distance
zbMATH Keywords: Monte Carlo results
zbMATH Keywords: risk improvement
reviewed by: Khursheed Alam
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176350705
OpenAlex ID: W2032795766
DOI: 10.1214/AOS/1176350705
description: scientific article; zbMATH DE number 4068053