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The Exact and Near-Exact Distributions for the Statistic Used to Test the Reality of Covariance Matrix in a Complex Normal Distribution

  • Luís M. GriloEmail author
  • Carlos A. Coelho
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 192)

Abstract

The authors start by approximating the exact distribution of the negative logarithm of the likelihood ratio statistic, used to test the reality of the covariance matrix in a certain complex multivariate normal distribution , by an infinite mixture of Generalized Near-Integer Gamma (GNIG) distributions. Based on this representation they develop a family of near-exact distributions for the likelihood ratio statistic, which are finite mixtures of GNIG distributions and match, by construction, some of the first exact moments. Using a proximity measure based on characteristic functions the authors illustrate the excellent properties of the near-exact distributions . They are very close to the exact distribution but far more manageable and have very good asymptotic properties both for increasing sample sizes as well as for increasing number of variables. These near-exact distributions are much more accurate than the asymptotic approximation considered, namely when the sample size is small and the number of variables involved is large. Furthermore, the corresponding cumulative distribution functions allow for an easy computation of very accurate near-exact quantiles.

Keywords

Characteristic function Beta distribution Gamma distribution Small samples Quantiles 

Notes

Acknowledgements

This work was partially supported by Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through project UID/MAT/00297/2013 (Centro de Matemática e Aplicações – CMA).

References

  1. 1.
    Anderson, T.W.: An Introduction to Multivariate Statistical Analysis, 3rd edn. Wiley, New York (2003)zbMATHGoogle Scholar
  2. 2.
    Berry, A.: The accuracy of the Gaussian approximation to the sum of independent variates. Trans. Am. Math. Soc. 49, 122–136 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brillinger, D.R.: Time Series: Data Analysis and Theory. SIAM, Philadelphia (2001)CrossRefzbMATHGoogle Scholar
  4. 4.
    Carter, E.M., Khatri, C.G., Srivastava, M.S.: Nonnull distribution of likelihood ratio criterion for reality of covariance matrix. J. Multivar. Anal. 6, 176–184 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Coelho, C.A.: The generalized integer gamma distribution - a basis for distributions in multivariate statistics. J. Multivar. Anal. 64, 86–102 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Coelho, C.A.: The generalized near-integer gamma distribution: a basis for “near-exact” approximations to the distribution of statistics which are the product of an odd number of independent beta random variables. J. Multivar. Anal. 89, 191–218 (2004)Google Scholar
  7. 7.
    Coelho, C.A.: Near-exact distributions: What are they and why do we need them? In: Proceedings 59th ISI World Statistics Congress, 25–30 August 2013, Hong Kong (Session STS084), pp. 2879–2884 (2013)Google Scholar
  8. 8.
    Coelho, C.A., Marques, F.J.: Near-exact distributions for the likelihood ratio test statistic to test equality of several variance-covariance matrices in elliptically contoured distributions. Commun. Stat. Theory Methods 27, 627–659 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Esseen, C.-G.: Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian Law. Acta Mathematica 77, 1–125 (1945)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gil-Pelaez, J.: Note on the inversion theorem. Biometrika 38, 481–482 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Goodman, N.R.: On the joint estimation of the spectra, cospectrum and quadrature spectrum of a two-dimensional stationary Gaussian process. Scientific Paper No. 10, Engineering Statistics Laboratory, New York University/Ph.D. Dissertation, Princeton University (1957)Google Scholar
  12. 12.
    Goodman, N.R.: Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction). Ann. Math. Stat. 34, 152–177 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Grilo, L.M.: Development of near-exact distributions for different scenarios of application of the Wilks Lambda statistic (in Portuguese). Ph.D. thesis, Lisbon University of Technology, Lisbon (2005)Google Scholar
  14. 14.
    Grilo, L.M., Coelho, C.A.: Development and comparative study of two near exact approximations to the distribution of the product of an odd number of independent beta random variables. J. Stat. Plan. Inference 137, 1560–1575 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Grilo, L.M., Coelho, C.A.: Near-exact distributions for the generalized Wilks Lambda statistic. Discuss. Math. Probab. Stat. 30, 53–86 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Grilo, L.M., Coelho, C.A.: The exact and near-exact distribution for the Wilks Lambda statistic used in the test of independence of two sets of variables. Am. J. Math. Manag. Sci. 30, 111–140 (2010)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Grilo, L.M., Coelho, C.A.: A family of near-exact distributions based on truncations of the exact distribution for the generalized Wilks Lambda statistic. Commun. Stat. Theory Methods 41, 2321–2341 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Grilo, L.M., Coelho, C.A.: Near-exact distributions for the likelihood ratio statistic used to test the reality of a covariance matrix. AIP Conf. Proc. 1558, 797–800 (2013)CrossRefGoogle Scholar
  19. 19.
    Hwang, H.-K.: On convergence rates in the central limit theorems for combinatorial structures. Eur. J. Comb. 19, 329–343 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    James, A.T.: Distributions of matrix variates and latent roots derived from normal samples. Ann. Math. Stat. 35, 475–501 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Johnson, N.L., Kotz, S., Balakrishnan, N.: Continuous Univariate Distributions, vol. 2. Wiley, New York (1995)zbMATHGoogle Scholar
  22. 22.
    Khatri, C.G.: Classical statistical analysis based on a certain multivariate complex Gaussian distribution. Ann. Math. Stat. 36, 98–114 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Khatri, C.G.: A test for reality of a covariance matrix in a certain complex Gaussian distribution. Ann. Math. Stat. 36, 115–119 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Marques, F.J., Coelho, C.A., Arnold, B.C.: A general near-exact distribution theory for the most common likelihood ratio test statistics used in multivariate analysis. TEST 20, 180–203 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Tricomi, F.G., Erdélyi, A.: The asymptotic expansion of a ratio of gamma functions. Pac. J. Math. 1, 133–142 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Wald, A., Brookner, R.J.: On the distribution of Wilk’s statistic for testing the independence of several groups of variates. Ann. Math. Stat. 12, 137–152 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Wooding, R.A.: The multivariate distribution of complex normal variables. Biometrika 43, 212–215 (1956)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Unidade Departamental de MatemáticaInstituto Politécnico de TomarTomarPortugal
  2. 2.Centro de Matemática e Aplicações (CMA/FCT-UNL)CaparicaPortugal
  3. 3.Departamento de Matemática (DM/FCT-UNL), Faculdade de Ciências e TecnologiaUniversidade Nova de LisboaCaparicaPortugal

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