Optimization of functions of matrices with an application in statistics
type: Item
instance of: scholarly article
title: Optimization of functions of matrices with an application in statistics
zbMATH Open document ID: 0841.15012
DOI: 10.1016/0024-3795(94)00087-5
author: Jean-Daniel Rolle
published in: Linear Algebra and its Applications
publication date: 1996-07-24T00:00:00Z
review text: Let \(M\) be a fixed \(N\times N\) non-zero idempotent symmetric matrix and consider the totality \(\mathcal P\) of \(R\times N\) matrices \(X\) such that \(XM\neq 0\). \(U\) is taken as an open subset of \(\mathcal P\), which is itself open in \(\mathbb{R}^{R\times N}\). The author calls an \((M; U)\) function a ratio of two linear functions of \(\text{tr}(MX' X)^2)\), \((\text{tr}(MX'X))^2\) and \(\text{tr}(MX' X)\). A theorem on finding global extrema of such functions through classical techniques using matrix differential calculus is given. Statistical application is made in the setting of the classical linear model \(\underset\widetilde{} Y= X\underset\widetilde{} \beta+ \underset\widetilde{} \varepsilon\), \(\underset\widetilde{} \varepsilon\sim {\mathcal N}(0, V)\), with the covariance matrix \(V\) permitted to be singular, and whose eigenvectors (but not eigenvalues) are assumed known. The problem addressed is the best (minimal MSE) estimation of the positive eigenvalues among classes of non-negative quadratic potential estimators. The article is in the tradition of \textit{J. R. Magnus} and \textit{H. Neudecker} [Matrix differential calculus with applications in statistics and econometrics (1988; Zbl 0651.15001)].
reviewed by: Eugene Seneta
Mathematics Subject Classification ID: 15A54
Mathematics Subject Classification ID: 62J05
Mathematics Subject Classification ID: 26B12
zbMATH DE Number: 847505
zbMATH Keywords: functions of matrices
zbMATH Keywords: linear regression model
zbMATH Keywords: global extrema
zbMATH Keywords: matrix differential calculus
zbMATH Keywords: positive eigenvalues
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)00087-5
OpenAlex ID: W2057196979
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description: scientific article; zbMATH DE number 847505