Approximation with a Kronecker product structure with one component as compound symmetry or autoregression via entropy loss function

Abstract The aim of this paper is to determine the best approximation of a positive definite symmetric matrix by a matrix having the Kronecker product structure with both components unstructured, or with one component structured as compound symmetry or the first-order autoregression, with the use of the entropy loss as a measure of discrepancy. We show some properties of the entropy loss function and we prove that in all the cases there exists the approximation with the positive definiteness property. Presented results can be widely used in multivariate statistics, for example for identification of the covariance structure of two-level multivariate data, or in testing hypotheses about the covariance structures. Simulation studies show that the proposed approach is reliable in the mentioned issues. Considered method is illustrated by the real data example.