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December 2008 Inference for eigenvalues and eigenvectors of Gaussian symmetric matrices
Armin Schwartzman, Walter F. Mascarenhas, Jonathan E. Taylor
Ann. Statist. 36(6): 2886-2919 (December 2008). DOI: 10.1214/08-AOS628


This article presents maximum likelihood estimators (MLEs) and log-likelihood ratio (LLR) tests for the eigenvalues and eigenvectors of Gaussian random symmetric matrices of arbitrary dimension, where the observations are independent repeated samples from one or two populations. These inference problems are relevant in the analysis of diffusion tensor imaging data and polarized cosmic background radiation data, where the observations are, respectively, 3×3 and 2×2 symmetric positive definite matrices. The parameter sets involved in the inference problems for eigenvalues and eigenvectors are subsets of Euclidean space that are either affine subspaces, embedded submanifolds that are invariant under orthogonal transformations or polyhedral convex cones. We show that for a class of sets that includes the ones considered in this paper, the MLEs of the mean parameter do not depend on the covariance parameters if and only if the covariance structure is orthogonally invariant. Closed-form expressions for the MLEs and the associated LLRs are derived for this covariance structure.


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Armin Schwartzman. Walter F. Mascarenhas. Jonathan E. Taylor. "Inference for eigenvalues and eigenvectors of Gaussian symmetric matrices." Ann. Statist. 36 (6) 2886 - 2919, December 2008.


Published: December 2008
First available in Project Euclid: 5 January 2009

zbMATH: 1196.62067
MathSciNet: MR2485016
Digital Object Identifier: 10.1214/08-AOS628

Primary: 62H12 , 62H15
Secondary: 62H11 , 92C55

Keywords: curved exponential family , likelihood ratio test , maximum likelihood , orthogonally invariant , Random matrix , submanifold

Rights: Copyright © 2008 Institute of Mathematical Statistics


Vol.36 • No. 6 • December 2008
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