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

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  • Carlos Coelho
  • Barry Arnold
  • Filipe Marques

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. Copyright Sociedad de Estadística e Investigación Operativa 2015

Suggested Citation

  • Carlos Coelho & Barry Arnold & Filipe Marques, 2015. "The exact and near-exact distributions of the main likelihood ratio test statistics used in the complex multivariate normal setting," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(2), pages 386-416, June.
    • Handle: RePEc:spr:testjl:v:24:y:2015:i:2:p:386-416
    DOI: 10.1007/s11749-014-0418-y
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    References listed on IDEAS

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    1. Carlos Coelho & Filipe Marques, 2012. "Near-exact distributions for the likelihood ratio test statistic to test equality of several variance-covariance matrices in elliptically contoured distributions," Computational Statistics, Springer, vol. 27(4), pages 627-659, December.
    2. Filipe Marques & Carlos Coelho & Barry Arnold, 2011. "A general near-exact distribution theory for the most common likelihood ratio test statistics used in Multivariate Analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(1), pages 180-203, May.
    3. Coelho, Carlos A., 1998. "The Generalized Integer Gamma Distribution--A Basis for Distributions in Multivariate Statistics," Journal of Multivariate Analysis, Elsevier, vol. 64(1), pages 86-102, January.
    4. Fang, C. & Krishnaiah, P. R. & Nagarsenker, B. N., 1982. "Asymptotic distributions of the likelihood ratio test statistics for covariance structures of the complex multivariate normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 12(4), pages 597-611, December.
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    Cited by:

    1. Thu, Pham-Gia & Phong, Duong Thanh, 2022. "The distribution of the non-central Wilks statistic in the complex case," Statistics & Probability Letters, Elsevier, vol. 184(C).

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