The Multi-Sample Block-Scalar Sphericity Test: Exact and Near-Exact Distributions for Its Likelihood Ratio Test Statistic

In this article the authors show how by adequately decomposing the null hypothesis of the multi-sample block-scalar sphericity test it is possible to obtain the likelihood ratio test statistic as well as a different look over its exact distribution. This enables the construction of well-performing near-exact approximations for the distribution of the test statistic, whose exact distribution is quite elaborate and non-manageable. The near-exact distributions obtained are manageable and perform much better than the available asymptotic distributions, even for small sample sizes, and they show a good asymptotic behavior for increasing sample sizes as well as for increasing number of variables and/or populations involved.