The exact and near-exact distributions of the main likelihood ratio test statistics used in the complex multivariate normal setting

The multi-sample independence test

“…where (x − μ) denotes the complex conjugate of (x − μ) and the prime denotes the transpose (see [6,8,12] for details). Let us consider that X k has a complex p-multivariate Normal distribution with expected value μ k and variancecovariance matrix Σ k , for k = 1,...,q and that we have q independent samples of dimension n, one from the distribution of each X k .…”

“…The likelihood ratio test statistics to test H 0a and H 0b|0a are given respectively by (see [6] for references)…”

“…Using the results in [6,10] is possible to show that the exact distribution of the likelihood ratio test statistic to test multi-sample independence, for an underlying multivariate Normal distribution, either real or complex, for the populations involved, has the distribution of the product of independent Beta distributions which has a complicated and non-manageable expression. To overcome this problem we propose the use of near-exact distributions [4,5,10] for the distribution of the test statistic and its logarithm.…”

“…These near exact distributions are very precise approximations which are obtained, in this case, by i) decomposing the null hypothesis of the test into two null hypotheses, one being the null hypothesis to test the equality of several Hermitian covariance matrices and the other the null hypothesis to test the independence of several groups of variables; ii) writing the characteristic function the logarithm of the likelihood ratio test statistic as, an adequate, product of characteristic functions; iii) and finally, leaving the major part of this characteristic function unchanged and asymptotically approximating the remaining part by another characteristic function in such a way that the resulting overall characteristic function corresponds to a well known distribution. In [6,10] the authors present a unified approach for developing near-exact distributions for the most common likelihood ratio test statistics in multivariate analysis, in particular the authors developed near-exact distributions for the test statistics used to test the independence of several groups of variables and the equality of several covariance matrices. In this paper we will address in detail the case where the samples are extracted from complex multivariate Normal populations.…”

“…Using a similar procedure it is also possible to deal with the real case. The relevance of studying the complex case is well described in [6] where the authors report that the complex multivariate Normal distribution arises in several areas of applications such as crystallography, spectral analysis in time series and performance of radar receivers [11,9,8,7].…”