Inference for eigenvalues and eigenvectors in exponential families of random symmetric matrices

Diffusion tensor imaging (DTI) data consist of a 3 ×3 positive definite random matrix at every voxel. Motivated by the anatomical interpretation of the data, we define a matrix-variate exponential family of distributions for random positive definite matrices and develop estimation and testing procedures for the eigenstructure of the mean parameter. The exponential family includes the spherical Gaussian and matrix-Gamma distributions as special cases. Maximum likelihood estimation and likelihood ratio testing procedures are carried out both in the one-sample and two-sample problems. In addition to their large-sample behavior, their non-asymptotic performance is evaluated via simulations. The methods are illustrated in a real DTI data example.