Convergence rates for empirical Bayes estimation in the uniform U(0,\(\theta\) ) distribution
type: Item
instance of: scholarly article
title: Convergence rates for empirical Bayes estimation in the uniform U(0,\(\theta\) ) distribution
zbMATH Open document ID: 0649.62003
author: Yoshiko Nogami
published in: The Annals of Statistics
publication date: 1988-01-01T00:00:00Z
review text: Let \(\{(X_ i,\theta_ i)\}\) be a sequence of independent random vectors where \(X_ i\) has a uniform density \(U(0,\theta_ i)\) for \(0<\theta_ i<m\) \((<\infty)\) and the unobservable \(\theta_ i\) are i.i.d. G in some class \({\mathcal G}\) of prior distributions. In the \((n+1)st\) problem we estimate \(\theta_{n+1}\) by \(t_ n(X_ 1,...,X_ n,X_{n+1})\doteq t_ n(X)\), incurring the risk \(R_ n\doteq E(t_ n(X)-\theta_{n+1})^ 2\), where E denotes expectation with respect to all random variables \(\{(X_ i,\theta_ i)\}^{n+1}_{i=1}\). Let R be the infimum Bayes risk with respect to G. In this paper the author exhibits empirical Bayes estimators with a convergence rate \(O(n^{-})\) of \(R_ n-R\) and shows that there is a sequence of empirical Bayes estimators for which \(R_ n-R\) has a lower bound of the same order \(n^{-}\).
Mathematics Subject Classification ID: 62C12
Mathematics Subject Classification ID: 62F10
Mathematics Subject Classification ID: 62C25
Mathematics Subject Classification ID: 62F15
zbMATH DE Number: 4060502
zbMATH Keywords: squared error loss
zbMATH Keywords: nonexponential family
zbMATH Keywords: uniform density
zbMATH Keywords: empirical Bayes estimators
zbMATH Keywords: convergence rate
zbMATH Keywords: lower bound
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176350966
OpenAlex ID: W2053536650
DOI: 10.1214/AOS/1176350966
Recommended article: Q3323021
Recommended article: Convergence Rates of Empirical Bayes Estimation for the Parameter of the Uniform DistributionU(0, θ) Under Random Censorship
Recommended article: Q3323023
Recommended article: Convergence rates for empirical Bayes two-action problems: The uniform \(U(0,{\theta{}})\) distribution
Recommended article: Empirical Bayes with rates and best rates of convergence in \(u(x)C(\Theta)exp(-x/{\Theta})\)-family: estimation case
Recommended article: Convergence rates for empirical Bayes estimation of binomial parameter
Recommended article: The convergent rate of empirical Bayes estimators for parameters of normal distribution family
description: scientific article; zbMATH DE number 4060502