Quadratic estimators of quadratic functions of normal parameters
type: Item
instance of: scholarly article
title: Quadratic estimators of quadratic functions of normal parameters
zbMATH Open document ID: 0609.62039
DOI: 10.1016/0378-3758(86)90105-9
published in: Journal of Statistical Planning and Inference
publication date: 1987-01-01T00:00:00Z
review text: When, for \(n\geq 2\), \(X_ 1,...,X_ n\) is a normal random sample with unknown mean \(\xi\) and unknown standard deviation \(\sigma\), the estimation problem of the form \(\theta =\xi^ 2+b\sigma^ 2\) under quadratic loss is considered, where b is a given non-negative number. The author discusses the admissibility of the estimator \[ \delta_ 0(\bar X,S)=\bar X^ 2+S^ 2(bn-1)\{n(n+1)\}^{-1} \] for the scaled quadratic loss \(L(\xi,\sigma,\delta)=(\theta -\delta)^ 2\sigma^{-4}\), under which \(\delta_ 0\) has a constant risk, where \(\bar X=\sum^{n}_{i=1}X_ i/n\) and \(S^ 2=\sum^{n}_{i=1}(X_ i-\bar X)^ 2\). In fact the estimator \(\delta_ 0\) is shown to be inadmissible in the case when \(n^{-1}\leq b<1+2n^{-1}\). It is also shown that \(\delta_ 0\) is admissible if and only if \(1+2n^{-1}\leq b<2+3n^{- 1}\), and that \(\delta_ 0\) is a generalized Bayes estimator with respect to a prior which admits a good approximation, in terms of posterior risk, by probability priors.
Mathematics Subject Classification ID: 62F10
Mathematics Subject Classification ID: 62C15
Mathematics Subject Classification ID: 62C20
zbMATH DE Number: 3986392
zbMATH Keywords: necessary and sufficient condition
zbMATH Keywords: square root transformation
zbMATH Keywords: normal parameters
zbMATH Keywords: quadratic loss
zbMATH Keywords: admissibility
zbMATH Keywords: scaled quadratic loss
zbMATH Keywords: generalized Bayes estimator
author: Andrew L. Rukhin
reviewed by: Masafumi Akahira
MaRDI profile type: Publication
full work available at URL: 0378-3758(86)90105-9
OpenAlex ID: W2025828373
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description: scientific article; zbMATH DE number 3986392