Estimation and testing of the commutative quadratic covariance structure under multivariate t distribution

Abstract This paper studies hypothesis testing for structured scale matrices in the multivariate t distribution. We develop likelihood ratio and Rao score tests for general structural constraints on the scale matrix and derive the Wald test for hypotheses in which the scale matrix is restricted to a quadratic subspace. The latter class includes, in particular, diagonality, sphericity, compound symmetry, circular Toeplitz structures, and their block extensions with structured subblocks. Since the construction of the test statistics requires maximum likelihood estimation, we derive the likelihood equations under quadratic subspace restrictions. Although likelihood ratio and score tests have been investigated for covariance structure in the multivariate normal model, the Wald test has not been developed in that setting. Because the multivariate normal distribution arises as a limiting case of the multivariate t model, our results also provide a new contribution to the normal framework. Finite sample performance is examined via simulation, and a real data example illustrates the proposed methodology.