On estimating a common multivariate normal mean vector
type: Item
instance of: scholarly article
title: On estimating a common multivariate normal mean vector
zbMATH Open document ID: 0602.62042
author: Wen-Jau Chiou
author: Arthur Cohen
published in: Annals of the Institute of Statistical Mathematics
publication date: 1985-01-01T00:00:00Z
review text: Let \(X_ 1,...,X_ n\) be a random sample of population \(N_ p(\mu,\Sigma_ X)\) and \(Y_ 1,...,Y_ n\) be a random sample of population \(N_ p(\mu,\Sigma_ Y)\). Assume the X-sample and Y-sample are independent. Consider the problem of estimating the common mean vector \(\mu\). When \(\Sigma_ X\) and \(\Sigma_ Y\) are known the unbiased estimator \[ T_ 3=\Sigma_ Y(\Sigma_ X+\Sigma_ Y)^{-1}\bar X+\Sigma_ X(\Sigma_ X+\Sigma_ Y)^{-1}\bar Y \] is better than \(\bar X\) and \(\bar Y\) in the sense that \(\Sigma_{\bar X}-\Sigma_{T_ 3}>0\) and \(\Sigma_{\bar Y}-\Sigma_{T_ 3}>0\). When \(\Sigma_ X\) and \(\Sigma_ Y\) are unknown it is logical to study the unbiased estimator \[ T_ 4=S_ Y(S_ X+S_ Y)^{-1}\bar X+S_ X(S_ X+S_ Y)^{- 1}\bar Y \] where \(S_ X\) and \(S_ Y\) are the sample covariance matrices of X and Y, respectively. In the paper the authors point out that neither \((\Sigma_{\bar X}-\Sigma_{T_ 4})\) nor \((\Sigma_{\bar Y}-\Sigma_{T_ 4})\) is positive semi-definite for all \((\Sigma_ X,\Sigma_ Y)\) for any n. They also give more general results of this negative type.
Mathematics Subject Classification ID: 62H12
zbMATH DE Number: 3971993
zbMATH Keywords: covariance matrix
zbMATH Keywords: unbiased estimators
zbMATH Keywords: multivariate normal distribution
zbMATH Keywords: mean vector
zbMATH Keywords: positive semi-definite
reviewed by: Kai-Tai Fang
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3971993