The multi-sample block-scalar sphericity test – exact and near-exact distributions for its likelihood ratio test statistic

In this paper the authors show how by adequately decomposing the null hypothesis of the multi-sample block sphericity test it is possible to easily obtain the expression for its likelihood ratio test statistic as well as a different look over the exact distribution of this statistic. This different view will enable the construction of very well-performing near-exact approximations for the distribution of this test statistic, whose exact distribution is quite elaborate and non-manageable. The near-exact distributions obtained are quite manageable and perform much better than the available asymptotic distributions, even for small sample sizes, and they show a good asymptotic behavior not only for increasing sample sizes as well as for increasing number of variables and/or samples involved.