Likelihood ratio test for covariance matrix under multivariate t distribution with uncorrelated observations

In this paper, estimators for the unknown parameters under two types of matrix-variate $t$ distributions are determined, and their basic statistical properties, including bias and sufficiency, are investigated. These estimators are then applied to test hypotheses concerning the covariance structure of a multivariate $t$ distribution associated with a collection of uncorrelated, though not necessarily independent, observation vectors, using two types of matrix-variate $t$ distributions. A likelihood ratio test is proposed, and its distributional properties under the null hypothesis are examined, assuming either a fully specified covariance matrix or one specified up to a constant. Furthermore, it is demonstrated that the asymptotic distribution for the type I matrix-variate $t$ distribution under both hypotheses coincides with that under the normality assumption. Finally, for testing a fully specified covariance matrix, the asymptotic distribution of the likelihood ratio test statistic is determined.