Improving on the positive part of the UMVUE of a noncentrality parameter of a noncentral chi-square distribution
type: Item
instance of: scholarly article
title: Improving on the positive part of the UMVUE of a noncentrality parameter of a noncentral chi-square distribution
zbMATH Open document ID: 0855.62040
author: Peter Yi-Shi Shao
author: William E. Strawderman
published in: Journal of Multivariate Analysis
publication date: 1997-02-02T00:00:00Z
review text: This paper gives explicit improvements to the positive part of the UMVUE of a noncentrality parameter of a noncentral \(\chi^2_n (\mu/2)\). Let \(Y \sim \chi^2_n (\mu/2)\) with \(n\) (known) degrees of freedom and unknown noncentrality parameter \(\mu\). We consider the problem of estimating \(\mu\) with loss \(= (\delta - \mu)^2\). The estimator \(Y + n\) is the generalized Bayes estimator of \(\mu\) with respect to a noninformative prior distribution and is dominated by \(Y - n\), the UMVUE of \(\mu\). Our results fall into two broad classes. The first set of results describes explicit estimators which change \((Y - n)_+\) only on the set \([n,n + 2]\). These results depend on some properties of central chi-square distributions which along with other preliminary results are given in Section 2. In Section 3, we will study alternative estimators of the form \[ \delta (a,g,Y) = (Y - n)_+ - ag(Y) I_{\{n \leq Y \leq n + 2\}}, \tag{1} \] where \(g (\cdot)\) is an even symmetric (``\(W\)-shaped'') piecewise linear function about \(Y = n + 1\) with \(g(n) = g(n + 2) = 0\), \(g'(n) < 0\) and \(|g'(Y) |\equiv 1\) a.e. on \([n,n + 2]\). To fully specify \(\delta (a,g,Y)\), it suffices to specify the constant \(a\) and a value \(y^*\) in \((n + 1,n + 2)\) such that \(g(y^*)\) attains its minimum. Values of \(a\) and \(y^*\) are given such that \(\delta (a,g,Y)\) dominates \((Y - n)_+\). In Section 4, we study estimators of the form (1) for certain nonpiecewise linear \(g\). It follows from the results of \textit{M. S. Chow} [Ann. Stat. 15, 800-804 (1987; Zbl 0627.62008)] that estimators of the form (1) cannot themselves be admissible because they are not generalized Bayes. Therefore we investigate a more general class of estimators in Section 5. This second class of estimators is of the form \[ \delta (Y) = (Y - n)_+ - ag (Y) I_{\{Y > q \}} - akh (Y) I_{\{0 \leq Y \leq q\}} \tag{2} \] with \(q > n\). We give conditions on \(a,k,q\), \(g (\cdot)\) such that (2) dominates \((Y …
Mathematics Subject Classification ID: 62H12
Mathematics Subject Classification ID: 62C99
zbMATH DE Number: 769826
zbMATH Keywords: noncentral chi-square distribution
zbMATH Keywords: noncentrality parameter
zbMATH Keywords: generalized Bayes estimator
zbMATH Keywords: positive part
zbMATH Keywords: UMVUE
zbMATH Keywords: central chi-square distributions
MaRDI profile type: Publication
full work available at URL: jmva.1995.1024
OpenAlex ID: W1997509514
DOI: 10.1006/JMVA.1995.1024
description: scientific article; zbMATH DE number 769826