{"title":"Truncated Estimators for a Precision Matrix","authors":"Anis M. Haddouche, Dominique Fourdrinier","doi":"10.3103/s1066530724700029","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In this paper, we estimate the precision matrix <span>\\({\\Sigma}^{-1}\\)</span> of a Gaussian multivariate linear regression model through its canonical form <span>\\(({Z}^{T},{U}^{T})^{T}\\)</span> where <span>\\(Z\\)</span> and <span>\\(U\\)</span> are respectively an <span>\\(m\\times p\\)</span> and an <span>\\(n\\times p\\)</span> matrices. This problem is addressed under the data-based loss function <span>\\(\\textrm{tr}\\ [({\\hat{\\Sigma}}^{-1}-{\\Sigma}^{-1})S]^{2}\\)</span>, where <span>\\({\\hat{\\Sigma}}^{-1}\\)</span> estimates <span>\\({\\Sigma}^{-1}\\)</span>, for any ordering of <span>\\(m,n\\)</span> and <span>\\(p\\)</span>, in a unified approach. We derive estimators which, besides the information contained in the sample covariance matrix <span>\\(S={U}^{T}U\\)</span>, use the information contained in the sample mean <span>\\(Z\\)</span>. We provide conditions for which these estimators improve over the usual estimators <span>\\(a{S}^{+}\\)</span> where <span>\\(a\\)</span> is a positive constant and <span>\\({S}^{+}\\)</span> is the Moore-Penrose inverse of <span>\\(S\\)</span>. Thanks to the role of <span>\\(Z\\)</span>, such estimators are also improved by their truncated version.</p>","PeriodicalId":46039,"journal":{"name":"Mathematical Methods of Statistics","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Methods of Statistics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3103/s1066530724700029","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we estimate the precision matrix \({\Sigma}^{-1}\) of a Gaussian multivariate linear regression model through its canonical form \(({Z}^{T},{U}^{T})^{T}\) where \(Z\) and \(U\) are respectively an \(m\times p\) and an \(n\times p\) matrices. This problem is addressed under the data-based loss function \(\textrm{tr}\ [({\hat{\Sigma}}^{-1}-{\Sigma}^{-1})S]^{2}\), where \({\hat{\Sigma}}^{-1}\) estimates \({\Sigma}^{-1}\), for any ordering of \(m,n\) and \(p\), in a unified approach. We derive estimators which, besides the information contained in the sample covariance matrix \(S={U}^{T}U\), use the information contained in the sample mean \(Z\). We provide conditions for which these estimators improve over the usual estimators \(a{S}^{+}\) where \(a\) is a positive constant and \({S}^{+}\) is the Moore-Penrose inverse of \(S\). Thanks to the role of \(Z\), such estimators are also improved by their truncated version.
期刊介绍：
Mathematical Methods of Statistics is an is an international peer reviewed journal dedicated to the mathematical foundations of statistical theory. It primarily publishes research papers with complete proofs and, occasionally, review papers on particular problems of statistics. Papers dealing with applications of statistics are also published if they contain new theoretical developments to the underlying statistical methods. The journal provides an outlet for research in advanced statistical methodology and for studies where such methodology is effectively used or which stimulate its further development.
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