Estimation and Testing Hypotheses in Two-Level and Three-Level Multivariate Data with Block Compound Symmetric Covariance Structure

This article deals with the estimation and hypotheses testing problems for two-level and three-level multivariate data. The coordinate-free approach is used to prove that the quadratic estimation of covariance parameters is equivalent to linear estimation with a properly defined inner product in the space of symmetric matrices for both two-level and three-level multivariate data. The estimators are shown to be unbiased, sufficient, complete, and consistent. A competitor for the likelihood ratio test on covariance components under linear constraints and the mean vectors are proposed, based on the F distribution. Simulation studies are conducted to see the power of the proposed tests and the proposed methods are implemented with two medical datasets.