Invariance properties of the likelihood ratio for covariance matrix estimation in some complex elliptically contoured distributions

The likelihood ratio (LR) for testing if the covariance matrix of the observation matrix X is R has some invariance properties that can be exploited for covariance matrix estimation purposes. More precisely, it was shown in Abramovich et al. (2004, 2007, 2007) that, in the Gaussian case, LR(R0|X), w...

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Bibliographic Details
Published inJournal of multivariate analysis Vol. 124; pp. 237 - 246
Main Authors Besson, Olivier, Abramovich, Yuri I.
Format Journal Article
LanguageEnglish
Published New York Elsevier Inc 01.02.2014
Taylor & Francis LLC
Elsevier
Subjects
Covariance matrix estimation
Elliptically contoured distribution
Estimating techniques
Expected likelihood
Likelihood ratio
Mathematical analysis
Mathematics
Matrix
Parameter estimation
Probability
Probability distribution
Studies
62H15
Likelihood ratio
Elliptically contoured distribution
62H10
Covariance matrix estimation
Expected likelihood
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Summary:The likelihood ratio (LR) for testing if the covariance matrix of the observation matrix X is R has some invariance properties that can be exploited for covariance matrix estimation purposes. More precisely, it was shown in Abramovich et al. (2004, 2007, 2007) that, in the Gaussian case, LR(R0|X), where R0 stands for the true covariance matrix of the observations X, has a distribution which does not depend on R0 but only on known parameters. This paved the way to the expected likelihood (EL) approach, which aims at assessing and possibly enhancing the quality of any covariance matrix estimate (CME) by comparing its LR to that of R0. Such invariance properties of LR(R0|X) were recently proven for a class of elliptically contoured distributions (ECD) in Abramovich and Besson (2013) and Besson and Abramovich (2013) where regularized CME were also presented. The aim of this paper is to derive the distribution of LR(R0|X) for other classes of ECD not covered yet, so as to make the EL approach feasible for a larger class of distributions. •We consider a class of complex elliptically contoured matrix distributions (ECD).•We investigate properties of the likelihood ratio (LR).•We derive stochastic representations of the LR for covariance matrix estimation (CME).•Its p.d.f. evaluated at the true CM R0 does not depend on the latter.•This extends the expected likelihood approach for regularized CME.
ISSN:0047-259X
1095-7243
DOI:10.1016/j.jmva.2013.10.024