The multi-sample block-scalar sphericity test under the complex multivariate Normal case

The multi-sample block-scalar sphericity test represents a wide family of tests since it has as particular cases several well known and fundamental tests in multivariate statistics, like for example the test of independence of several groups of variables, the sphericity test and the test of equality of several variance-covariance matrices. In this work we address the case where we have several independent samples extracted from complex multivariate Normal populations and we want to test if the covariance matrices are equal for all populations and if for each population they have a particular diagonal structure. We show that by decomposing the null hypothesis into three partial null hypotheses it is possible to easily derive the likelihood ratio test statistic, the expression of itsh-th moment and thecharacteristicfunction of its logarithm. Using this method we are able to betteranalyzethe structure of the exact distribution and, using the induced factorization on thecharacteristicfunction, we show how it is possible to develop simple but highly accurate near-exact distributions for the likelihood ratio test statistic.