Near-exact distributions for the independence and sphericity likelihood ratio test statistics

In this paper we will show how a suitably developed decomposition of the characteristic function of the logarithm of the likelihood ratio test statistic to test independence in a set of variates may be used to obtain extremely wellt near-exact distributions both for this test statistic as well as for the likelihood ratio test statistic for sphericity, based on a decomposition of this latter test in two independent tests. For the independence test statistic, numerical studies and comparisons with asymptotic distributions proposed by other authors show the extremely high closeness of the near-exact distributions developed to the exact distribution. Concerning the sphericity test statistic, comparisons with the near-exact distributions developed in [19] show the advantages of these new near-exact distributions.