Speaker: Armin Schwartzman, Department of Biostatistics, Harvard School
of Public Health and Dana-Farber Cancer Institute
 
Title: Inference for Eigenvalues and Eigenvectors of Gaussian Symmetric 
Matrices
 
Abstract:
 
This work 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, where the observations are 3-by-3
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 here, 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.