# Near-Exact Distributions for the Likelihood Ratio Test Statistic for Testing Multisample Independence—The Real and Complex Cases

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## Abstract

We consider a generalization of the well-known independence of several groups of variables test, which we designate by multisample independence test of several groups of variables. This new generalization is of great interest whenever we want to test if in different populations, which may follow a multivariate complex or real normal distribution, the Hermitian covariance matrices have the same structure and if there is independence between different groups of variables. We show that the test statistic has the distribution of the product of independent beta random variables; however, the explicit expressions for the probability density and cumulative distribution functions turn out to be very complicated and almost impossible to use in practice. Our objective is to use a breakthrough technique to develop near-exact distributions for the test statistic. These approximations are known to be highly accurate and easy to use, which facilitates and encourages their use in practice. Using a decomposition of the null hypothesis of the test into two null hypotheses we obtain, in a simple way, the likelihood ratio test statistic, the expression of its *h*th null moment, and the characteristic function of its logarithm. The decomposition of the null hypothesis also induces a factorization on the characteristic function of the logarithm of the test statistic, which enables the development of near-exact distributions. The numerical studies presented highlight the good properties of these approximations and show their great precision. Simulation studies conducted show the good power of the test proposed even for alternatives quite close to the null hypothesis. An example of application of the test is also provided.

## Keywords

Equality of covariance matrices test Generalized near-integer gamma distribution Independence of several groups of variables test Mixtures Multivariate complex normal distribution## AMS Subject Classification

62H10 62E20 62H05 62H15## Preview

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