Testing simultaneously different covariance block diagonal structures - the multi-sample case

In this work a likelihood ratio test which allows to test simultaneously if, several covariance matrices are equal and block diagonal with different specific structures in the diagonal blocks, is developed. The distribution of the likelihood ratio statistic is studied and the expression of its hth null moment are derived. In order to make the test useful in practical terms, near-exact approximations are developed for the likelihood ratio statistic. A practical application to real data set together with numerical studies and simulations are provided in order illustrate the applicability of the test and also to assess the precision of the near-exact approximations developed.