A sharp necessary and sufficient condition for inadmissibility of estimators in a control problem
type: Item
instance of: scholarly article
title: A sharp necessary and sufficient condition for inadmissibility of estimators in a control problem
zbMATH Open document ID: 0547.62006
author: N. E. Zubov
published in: The Annals of Statistics
publication date: 1984-01-01T00:00:00Z
review text: Let a standard normal model of the control problem \(Y=\theta^ tz+\epsilon\) be given, where \(\theta\) has to be estimated with respect to the loss \(L(\theta,\delta)=(1-\theta^ t\delta)^ 2\). An estimator for \(\delta\) can be evaluated by its risk function \(R(\theta\),\(\delta)\), and is called inadmissible if there exists \(\delta^*\) such that \(R(\theta,\delta^*)\leq R(\theta,\delta)\) for all \(\theta\) with strict inequality for some \(\theta\). This paper deals with orthogonally invariant nonrandomized decision rules of the form \(\delta (x)=\phi (| x|)x/| x|\). Generalized Bayes estimators with respect to spherically symmetric priors have such a representation. After three preparatory lemmas in section 2, a necessary condition (theorem 3.1) and a sufficient condition (theorem 4.1) for the mentioned generalized Bayes estimators are shown. Applications (theorems 5.1, 5.2) and comparisons with different approaches of the admissibility concept [\textit{J. O. Berger, L. M. Berliner} and \textit{A. Zaman,} ibid. 10, 838-856 (1982; Zbl 0496.62005)] are given in section 5.
Mathematics Subject Classification ID: 62C15
Mathematics Subject Classification ID: 62F15
Mathematics Subject Classification ID: 62F10
Mathematics Subject Classification ID: 62P20
zbMATH DE Number: 3872461
zbMATH Keywords: inadmissibility
zbMATH Keywords: formal Bayes estimators
zbMATH Keywords: control problem
zbMATH Keywords: orthogonally invariant nonrandomized decision rules
zbMATH Keywords: spherically symmetric priors
zbMATH Keywords: necessary condition
zbMATH Keywords: sufficient condition
zbMATH Keywords: generalized Bayes estimators
zbMATH Keywords: Applications
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176346712
OpenAlex ID: W1995931214
DOI: 10.1214/AOS/1176346712
description: scientific article; zbMATH DE number 3872461