Approximation with a Kronecker product structure with one component as compound symmetry or autoregression

The aim of this paper is to determine the Kronecker product of two matrices, where the first component is structured as compound symmetry or autoregression of order one and the second one is arbitrary, such that the Frobenius norm of this matrix and a given, arbitrary matrix (unstructured or separable) is minimized. The properties of this nearest approximation, such as non-negativity, symmetry or positive definiteness, are shown. The proposed method can be used for example in statistical research: for regularizing the covariance structure of a given covariance matrix, for determining the estimators of covariance structure or for testing hypotheses about the covariance structures. Simulation studies show that the proposed approach is reliable in the mentioned issues.