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

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## Abstract

In this paper the authors show how it is possible to establish a common structure for the exact distribution of the main likelihood ratio test (LRT) statistics used in the complex multivariate normal setting. In contrast to what happens when dealing with real random variables, for complex random variables it is shown that it is possible to obtain closed-form expressions for the exact distributions of the LRT statistics to test independence, equality of mean vectors and the equality of an expected value matrix to a given matrix. For the LRT statistics to test sphericity and the equality of covariance matrices, cases where the exact distribution has a non-manageable expression, easy to implement and very accurate near-exact distributions are developed. Numerical studies show how these near-exact distributions outperform by far any other available approximations. As an example of application of the results obtained, the authors develop a near-exact approximation for the distribution of the LRT statistic to test the equality of several complex normal distributions.

## Keywords

Covariance matrix Equality of covariance matrices Equality of mean vectors Fourier transforms Generalized integer gamma (GIG) distribution Generalized near-integer gamma (GNIG) distribution Independence Mixtures Expected value matrix Sphericity Statistical distributions (distribution functions)## Mathematics Subject Classification

62H05 62H10 62E15 62H15 62E20## Notes

### Acknowledgments

Research partially supported by National Funds through FCT-Fundação para a Ciência e a Tecnologia, project PEst-OE/MAT/UI0297/2014 (CMA/UNL). The authors would like to express their gratitude to two anonymous Referees, the Associate Editor and also the Editor-in-Chief, whose suggestions contributed to a more solid and self-contained paper.

## Supplementary material

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