## 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.

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## 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.

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## Appendix A: Short user’s guide for the computational modules

### Appendix A: Short user’s guide for the computational modules

### A.1 Modules for the exact distributions of statistics in Sect. 2

Concerning the modules for the exact distributions of the three LRT statistics addressed in Sect. 2, there are five modules made available for each statistic \(\varLambda _i\). These modules are CFWi, PDFWi, CDFWi, PDFLi and CDFLi, where \(i\) is to be replaced by either 1, 2 or 3. These modules give as output, respectively, the value for the c.f. of \(W_i=-\log \,\varLambda _i\) \({(i=1,2,3)}\), the value for its pdf, the value for its cdf, the value for the pdf of \(\varLambda _i\) and the value for the cdf of \(\varLambda _i\), computed at the value for the running variable.

In the case of the LRT statistic \(\varLambda _1\), the arguments of these five modules are, in this order: (i) a list whose components are the number of variables of each group, (ii) the sample size, (iii) the running variable for the function. For \(\varLambda _2\), the arguments are, in this order: (i) the number of variables (equal for each population), (ii) the number of mean vectors being tested, (iii) the overall sample size, (iv) the running variable for the function. In the case of \(\varLambda _3\), the arguments are (i) the number of variables, (ii) either the rank of the matrix \(D\), in case this matrix is used, or the number of columns of the matrix \(\mu \).

The modules used to compute the pdf and cdf of \(W_i=-\log \varLambda _i\) make use of the modules for the pdf and cdf of the GIG distribution, while the modules for the pdf and cdf of \(\varLambda _i\) make use of the corresponding modules for \(W_i\), using the usual technique of r.v. transformation.

To obtain the computational modules and for a few examples of their use download the Mathematica\(^{{\circledR }}\) file Mathematica_Modules_for_Exact_distr_stats_Sec_2.nb, readily available from the web-page https://sites.google.com/site/nearexactdistributions/complex-normal.

### A.2 Modules for the near-exact distributions of statistics in Sects. 3 and 4

For the statistics \(\varLambda _4\) and \(\varLambda _5\), in Sects. 3 and 4 and whose near-exact distributions are developed in Sect. 5, there are also five modules made available for each statistic. These modules are: NECFWi, NEPDFWi, NECDFWi, NEPDFLi and NECDFLi, where \(i\) is to be replaced by either 4 or 5. These modules give as output, respectively, the value for the near-exact c.f. of \(W_i=-\log \varLambda _i\) \({(i=4,5)}\), the value for its near-exact pdf and cdf and the value for the near-exact pdf and cdf of \(\varLambda _i\), computed at the value for the running variable.

In the case of the LRT statistic \(\varLambda _4\), the arguments of the five modules are, in the following order: (i) the number of variables, (ii) the sample size, (iii) the number of exact moments to be matched, (iv) the running variable for the function and (v) an optional argument which is the number of precision digits used to compute the exact moments to be matched by the near-exact distribution and which has a default value of 100. For \(\varLambda _5\), the arguments are, in the following order: (i) the number of variables, (ii) the number of covariance matrices being tested, (iii) the sample size, (iv) the number of exact moments to be matched, (v) the running variable for the function and (vi) the same optional argument as in (v) for \(\varLambda _4\).

The modules used to compute the near-exact pdf and cdf of \(W_i=-\log \,\varLambda _i\) \({(i=4,5)}\) make use of the modules for the pdf and cdf of the GIG and GNIG distributions, while the modules for the near-exact pdf and cdf of \(\varLambda _i\) make use of the corresponding modules for \(W_i\), by using the usual technique of r.v. transformation.

To obtain the computational modules and for a few examples of their use download the Mathematica\(^{{\circledR }}\) file Mathematica_Modules_for_Near-Exact_distr_stats_Sec_3_4.nb, readily available from the web-page https://sites.google.com/site/nearexactdistributions/complex-normal.

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Coelho, C.A., Arnold, B.C. & Marques, F.J. The exact and near-exact distributions of the main likelihood ratio test statistics used in the complex multivariate normal setting.
*TEST* **24, **386–416 (2015). https://doi.org/10.1007/s11749-014-0418-y

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### 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