Two-stage sequential estimation of a multivariate normal mean under quadratic loss
type: Item
instance of: scholarly article
title: Two-stage sequential estimation of a multivariate normal mean under quadratic loss
zbMATH Open document ID: 0591.62072
author: Jayalakshmi Natarajan
author: William E. Strawderman
published in: The Annals of Statistics
publication date: 1985-01-01T00:00:00Z
review text: Suppose \(Y_1,Y_2,\ldots,Y_ n\) are \(p\)-dimensional i.i.d. \(N(\theta,I)\) where the unknown \(\theta\) is to be estimated under quadratic loss \(L(\theta,\delta) = \|\delta-\theta\|^2\). The authors search two-stage sequential estimators which are better both in risk and sample size than the usual estimator of a given sample size \(n\). Let \(\bar X_ i = \sum^{i}_{j=1} Y_ j/i\). Stopping rules based on \(\|\bar X_ m\|\), where \(1\leq m\leq n-1\), and estimators of the form \[ \delta^ m_ c (Y_1, \ldots,Y_ n) = \begin{cases} \bar X_ m, \quad&\text{if \(\|\bar X_ m\| <c\),}\\ (1/an^{-1}\|\bar X_ n\|^{-2})\bar X_ n, \quad&\text{otherwise,} \end{cases} \] where \(0<a<2(p-2)\) are considered. It is shown that for \(p\geq3\), \(n\geq2\) there exists \(c^ m(a)>0\) such that for all \(0<c\leq c^ m(a)\) the estimators \(\delta^ m_ c\) have lower risks then \(\bar X_ n.\) Modifications of \(\delta^ m_ c\) replacing \(\bar X_ m\) by the corresponding James-Stein estimators and the case of unknown covariance matrix are also considered.
Mathematics Subject Classification ID: 62L12
Mathematics Subject Classification ID: 62H12
Mathematics Subject Classification ID: 62C15
Mathematics Subject Classification ID: 62F10
Mathematics Subject Classification ID: 62L99
zbMATH DE Number: 3949537
zbMATH Keywords: multivariate normal mean
zbMATH Keywords: quadratic loss
zbMATH Keywords: two-stage sequential estimators
zbMATH Keywords: James-Stein estimators
zbMATH Keywords: unknown covariance matrix
reviewed by: Krzysztof J. Szajowski
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176349752
OpenAlex ID: W2026389891
DOI: 10.1214/AOS/1176349752
description: scientific article; zbMATH DE number 3949537