The Multi-Sample Independence Test

The multi-sample independence test for several groups of variables is a generalization of the usual independencetest which is of great importance in several areas of application. The exact distribution of the likelihood ratio test statistic ofthis test, both in the real or complex multivariate Normal setting, has a non-manageable and complicated expression. Usinga novel approach, we develop near-exact distributions for the distribution of the test statistic which are highly accurate andeasy to use. These are obtained decomposing the null hypothesis of the test into two null hypotheses, one to test the equalityof the several variance-covariance matrices and the other to test, assuming that the first null hypothesis is not rejected, theindependence of several groups of variables. This procedure allows us to obtain, in a simple way, the likelihood ratio teststatistic, the expression of its h-th null moment and the characteristic function of its logarithm. The decomposition of thenull hypothesis induces a factorization of the characteristic function of the logarithm of the test statistic which enables thedevelopment of near-exact distributions. The near-exact distributions obtained have the form of mixtures of Generalized Near-Integer Gamma distributions revealing a hight degree of precision in the approximation and good asymptotic properties forthe different parameters involved.