Estimation of the inverse scatter matrix of an elliptically symmetric distribution

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• 作者：Dominique Fourdrinier^a ; ^{Dominique.Fourdrinier@univ-rouen.fr" class="auth_mail" title="E-mail the corresponding author} ; Fatiha Mezoued^b ; ^{mezoued.fatiha@enssea.dz" class="auth_mail" title="E-mail the corresponding author} ; Martin T. Wells^c ; ^{mtw1@cornell.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：62H12 ; 62F10 ; 62C99
• 刊名：Journal of Multivariate Analysis
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：143
• 期：Complete
• 页码：32-55
• 全文大小：501 K

文摘

We consider estimation of the inverse scatter matrices Σ⁻¹${\Sigma }^{-1}$ for high-dimensional elliptically symmetric distributions. In high-dimensional settings the sample covariance matrix S$S$ may be singular. Depending on the singularity of S$S$, natural estimators of Σ⁻¹${\Sigma }^{-1}$ are of the form $a{S}^{-1}$ or $a{S}^{+}$ where a$a$ is a positive constant and S⁻¹$S^{-1}$ and S⁺$S^{+}$ are, respectively, the inverse and the Moore–Penrose inverse of S$S$. We propose a unified estimation approach for these two cases and provide improved estimators under the quadratic loss $\mathrm{tr}{({\hat{\Sigma }}^{-1}-{\Sigma }^{-1})}^2$. To this end, a new and general Stein–Haff identity is derived for the high-dimensional elliptically symmetric distribution setting.