Likelihood Ratio Test for the Hyper- Block Matrix Sphericity Covariance Structure – Characterization of the Exact Distribution and Development of Near-exact Distributions for the Test Statistic

• In this paper the authors introduce the hyper-block matrix sphericity test which is a generalization of both the block-matrix and the block-scalar sphericity tests and as such also of the common sphericity test. This test is a tool of crucial importance to verify elaborate assumptions on covariance matrix structures, namely on metaanalysis and error covariance structures in mixed models and models for longitudinal data. The authors show how by adequately decomposing the null hypothesis of the hyper-block matrix sphericity test it is possible to easily obtain the expression for the likelihood ratio test statistic as well as the expression for its moments. From the factorization of the exact characteristic function of the logarithm of the statistic, induced by the decomposition of the null hypothesis, and by adequately replacing some of the factors with an asymptotic result, it is possible to obtain near-exact distributions that lie very close to the exact distribution. The performance of these near-exact distributions is assessed through the use of a measure of proximity between distributions, based on the corresponding characteristic functions. Key-Words: • Equality of matrices test, Generalized Integer Gamma distribution, Generalized NearInteger Gamma distribution, Independence test, Near-exact distributions, Mixtures of distributions AMS Subject Classification: • 62H15, 62H10, 62E10, 62E15, 62E20. 2 Bárbara R. Correia, Carlos A. Coelho and Filipe J. Marques The hyper-block matrix sphericity test 3