The advantage of decomposing elaborate hypotheses on covariance matrices into conditionally independent hypotheses in building near-exact distributions for the test statistics

The aim of this paper is to show how the decomposition of elaborate hypotheses on the structure of covariance matrices into conditionally independent simpler hypotheses, by inducing the factorization of the overall test statistic into a product of several independent simpler test statistics, may be used to obtain near-exact distributions for the overall test statistics, even in situations where asymptotic distributions are not available in the literature and adequately fit ones are not easy to obtain.