The Annals of Statistics

Inference for eigenvalues and eigenvectors of Gaussian symmetric matrices

Armin Schwartzman, Walter F. Mascarenhas, and Jonathan E. Taylor

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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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Ann. Statist., Volume 36, Number 6 (2008), 2886-2919.

First available in Project Euclid: 5 January 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H15: Hypothesis testing 62H12: Estimation
Secondary: 62H11: Directional data; spatial statistics 92C55: Biomedical imaging and signal processing [See also 44A12, 65R10, 94A08, 94A12]

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


Schwartzman, Armin; Mascarenhas, Walter F.; Taylor, Jonathan E. Inference for eigenvalues and eigenvectors of Gaussian symmetric matrices. Ann. Statist. 36 (2008), no. 6, 2886--2919. doi:10.1214/08-AOS628.

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