Near-exact distributions for certain likelihood ratio test statistics

In this paper we will show how, using an expansion of a Logbeta distribution as an infinite mixture of Gamma distributions we are able to obtain near-exact distributions for the negative logarithm of the likelihood ratio test statistics used in Multivariate Analysis to test the independence of several sets of variables, the equality of several mean vectors, sphericity and the equality of several variance-covariance matrices as finite mixtures of Generalized Near-Integer Gamma distributions. These near-exact distributions will match as many of the exact moments as we wish and we will be able to have an {\emph a priori} upper-bound for the difference between their c.d.f.\ and the exact c.d.f.. These near-exact distributions also display very good performances, with an agreement with the exact distribution which may virtually be taken as far as we wish and which is by no means possible to obtain with the usual asymptotic distributions.