How to Develop Near-Exact Distributions for Likelihood Ratio Statistics used to Test Different Covariance Structures

The analysis of the covariance structure is a crucial topic in multivariate statistics. Not only does the covariance matrix contain valuable information about the dependence structure between variables, enabling informed conclusions and optimal decision-making, but it also helps in calibrating statistical models. With the increasing use of more complex models, it has become essential to verify assumptions about the structure of covariance matrices. In this study, we review several significant covariance structures and demonstrate how it is possible to simultaneously test the presence of different structures in the diagonal blocks of a covariance matrix. We examine the distribution of the likelihood ratio test statistic and derive the expression for its h-th null moment.

To ensure practical usability, we develop near-exact approximations for the likelihood ratio statistic. We provide a practical application using real data, along with numerical studies and simulations, to illustrate the applicability of the test and assess the precision of the developed near-exact approximations. Finally, we demonstrate how the proposed methodology can be extended to the study of more complex covariance structures.