Near-exact distributions for the likelihood ratio statistic used to test the reality of a covariance matrix

In a first stage the authors show how the exact distribution of the likelihood ratio statistic used to test the reality of a covariance matrix may be expressed as the distribution of the sum of two independent random variables, one with a Generalized Integer Gamma distribution and the other with the distribution of a sum of independent Logbeta random variables. From this form of the exact distribution the authors develop then a family of near-exact distributions, based on finite mixtures of Generalized Near-Integer Gamma distributions. These near-exact distributions match, by construction, some of the first exact moments and they have very manageable cumulative distribution functions, which allow for an easy computation of sharp near-exact quantiles and p-values.