Near-exact approximations for the likelihood ratio test statistic for testing equality of several variance-covariance matrices

While on one hand, the exact distribution of the likelihood ratio test statistic for testing the equality of several variance-covariance matrices, as it also happens with several other likelihood ratio test statistics used in Multivariate Statistics, has a non-manageable form, which does not allow for the computation of quantiles, even for a small number of variables, on the other hand the asymptotic approximations available do not have the necessary quality for small sample sizes. This way, the development of near-exact approximations to the distribution of this statistic is a good goal. Starting from a factorization of the exact characteristic function for the statistic under study and by adequately replacing some of the factors, we obtain a near-exact characteristic function which determines the near-exact distribution for the statistic. This near-exact distribution takes the form of either a GNIG (Generalized Near-Integer Gamma) distribution or a mixture of GNIG distributions. The evaluation of the performance of the near-exact and asymptotic distributions developed is done through the use of two measures based on the charactersitic function with which we are able to obtain good upper-bounds on the absolute value of the difference between the exact and approximate probability density or cumulative distribution functions. As a reference we use the asymptotic distribution proposed by Box.