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 l.r.t. (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 which will match as many of the exact moments as we wish and for which we will be able to have an 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 performance, with an agreement with the exact distribution which may virtually be taken as far as we wish and which it is not possible to obtain with the usual asymptotic distributions. Furthermore, based on the results presented it will be easy to build near-exact distributions for any l.r.t. statistics which may be built as the product of the above l.r.t. statistics.