Wishart and chi-square distributions associated with matrix quadratic forms
type: Item
instance of: scholarly article
title: Wishart and chi-square distributions associated with matrix quadratic forms
zbMATH Open document ID: 0948.62033
author: Thomas Mathew
author: Kenneth Nordström
published in: Journal of Multivariate Analysis
publication date: 2000-11-06T00:00:00Z
review text: The authors provide necessary and sufficient conditions for the distributional form of certain matrix quadratic forms. Specifically, they considered an \(n\times p\) matrix \(\mathbf Y\), which follows the normal distribution with parameters 0, \(\Sigma_{\mathbf Y}\) and assuming that \(\mathbf Q\) is a nonnegative definite \(n\times n\) matrix, they obtained necessary and sufficient conditions under which (i) the matrix \({\mathbf Y}'{\mathbf{QY}}\) follows some Wishart distribution (without requiring knowledge of the specific values of its parameters, say \(m\) and \(\Sigma\)) and (ii) the distribution of the random variable \(\ell'{\mathbf Y}'{\mathbf{QY}}\ell\) is a multiple of a chi-square distribution for every \(p\times 1\) vector \(\ell\). For the first problem, they have proved that \({\mathbf Y}'{\mathbf{QY}}\) follows a Wishart distribution if and only if one of the following equivalent conditions holds: (i) \(({\mathbf T}'\otimes {\mathbf I}_{\mathbf p})\Sigma_{\mathbf Y}({\mathbf T}\otimes{\mathbf I}_{\mathbf p})={\mathbf C}\otimes \Lambda\), where \({\mathbf T}\) is an \(n\times r\) matrix of rank \(r\), \(m{\mathbf C}\) is a symmetric and idempotent \(r\times r\) matrix with \(m= \text{rank}({\mathbf C})\) and \(\Lambda= E({\mathbf Y}'{\mathbf {QY}})\), (ii) \(({\mathbf Q}\otimes{\mathbf I}_{\mathbf p}) \Sigma_{\mathbf Y} ({\mathbf Q}\otimes{\mathbf I}_{\mathbf p})=m^{-1} {\mathbf{QAQ}}\otimes \Lambda\), where \({\mathbf A}\) is a nonnegative definite \(n\times n\) matrix such that \({\mathbf {QA}}\) is idempotent and \(m=\text{rank}({\mathbf{QA}})\). If one of these conditions holds then the distribution of \({\mathbf Y}'{\mathbf{QY}}\) is \(W_p (m,\Sigma)\), with \(\Sigma= m^{-1}\Lambda\). As the authors point out, these conditions can be verified effortlessly and they do not require a predetermination of the parameters of the Wishart distribution -- something which was essential for the verification of some similar conditions that were proposed ear…
Mathematics Subject Classification ID: 62H05
Mathematics Subject Classification ID: 62H10
zbMATH DE Number: 1053246
zbMATH Keywords: complex covariance structure
zbMATH Keywords: group symmetry covariance model
zbMATH Keywords: multivariate components of variance model
zbMATH Keywords: skew-symmetric matrix
zbMATH Keywords: chi-square distribution
zbMATH Keywords: Wishart distribution
reviewed by: Evdokia Xekalaki
MaRDI profile type: Publication
full work available at URL: jmva.1997.1665
OpenAlex ID: W2150614277
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DOI: 10.1006/JMVA.1997.1665
description: scientific article; zbMATH DE number 1053246