Inference for eigenvalues and eigenvectors of Gaussian symmetric matrices

TL;DR

This paper develops maximum likelihood estimators and likelihood ratio tests for the eigenvalues and eigenvectors of Gaussian symmetric matrices, with applications in imaging and cosmic data analysis.

Contribution

It introduces closed-form MLEs and LLRs for eigenparameters under orthogonally invariant covariance structures, extending inference methods for symmetric matrices.

Findings

MLEs do not depend on covariance parameters under orthogonal invariance

Closed-form expressions for MLEs and LLRs are derived

Applicable to diffusion tensor imaging and cosmic background data

Abstract

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, $3\times3$ and $2\times2$ 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.