Likelihood Ratio Tests for Elaborate Covariance Structures and for MANOVA Models with Elaborate Covariance Structures-A Review

In this paper a review is made from the primordia of the history of likelihood ratio tests for covariance structures and equality of mean vectors through the development of likelihood ratio tests that refer to elaborate covariance structures. Relations are established among several covariance structures, taking more elaborate ones as umbrella structures and examining then their particular cases of interest. References are made to bibliography where the corresponding likelihood ratio tests are developed and the distributions of the corresponding statistics addressed. Most of the likelihood ratio test statistics for one-way MANOVA models where the covariance matrices have elaborate structures were developed quite recently. Also for these likelihood ratio tests a similar approach is taken. Although we start with the common test that uses unstructured covariance matrices, then we go on to consider tests with more elaborate covariance structures, and subsequently we specify them to their particular cases of interest. Some special attention is also given to the so-called Wilks A statistics.