Near-exact distributions for the likelihood ratio test statistic of the multi-sample block-matrix sphericity test

The multi-sample block-matrix sphericity test and its particular cases have wide applications in different areas of research, as for example to test the error structure in several multivariate linear models.However, the practical implementation of this test has been hindered by difficulties in handling the exact distribution of the associated statistic and the non-availability in the literature of well-fitting asymptotic distributions.We use a decomposition of the null hypothesis into three null hypotheses which will induce a factorization in the likelihood ratio test (l.r.t.) statistic. We thenuse the induced factorization of the characteristic function (c.f.) of the logarithm of the l.r.t. statistic to obtain very well-fitting and highly manageable near-exact distributions for the l.r.t. statistic of this test and its particular cases.These near-exact distributions will allow for the easy computation of very accurate near-exact quantiles and $p$-values, enabling this way a more frequent practical use of these tests. A measure of proximity between distributions, based on the corresponding characteristic functions, is used to assess the performance of the near-exact distributions.