The Exact and Near-Exact Distributions for the Statistic Used to Test the Reality of Covariance Matrix in a Complex Normal Distribution

Abstract

The authors start by approximating the exact distribution of the negative logarithm of the likelihood ratio statistic, used to test the reality of the covariance matrix in a certain complex multivariate normal distribution , by an infinite mixture of Generalized Near-Integer Gamma (GNIG) distributions. Based on this representation they develop a family of near-exact distributions for the likelihood ratio statistic, which are finite mixtures of GNIG distributions and match, by construction, some of the first exact moments. Using a proximity measure based on characteristic functions the authors illustrate the excellent properties of the near-exact distributions . They are very close to the exact distribution but far more manageable and have very good asymptotic properties both for increasing sample sizes as well as for increasing number of variables. These near-exact distributions are much more accurate than the asymptotic approximation considered, namely when the sample size is small and the number of variables involved is large. Furthermore, the corresponding cumulative distribution functions allow for an easy computation of very accurate near-exact quantiles.