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Near-exact distributions for the independence and sphericity likelihood ratio test statistics

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  • Coelho, Carlos A.
  • Marques, Filipe J.

Abstract

In this paper we show how, based on a decomposition of the likelihood ratio test for sphericity into two independent tests and a suitably developed decomposition of the characteristic function of the logarithm of the likelihood ratio test statistic to test independence in a set of variates, we may obtain extremely well-fitting near-exact distributions for both test statistics. Since both test statistics have the distribution of the product of independent Beta random variables, it is possible to obtain near-exact distributions for both statistics in the form of Generalized Near-Integer Gamma distributions or mixtures of these distributions. For the independence test statistic, numerical studies and comparisons with asymptotic distributions proposed by other authors show the extremely high accuracy of the near-exact distributions developed as approximations to the exact distribution. Concerning the sphericity test statistic, comparisons with formerly developed near-exact distributions show the advantages of these new near-exact distributions.

Suggested Citation

  • Coelho, Carlos A. & Marques, Filipe J., 2010. "Near-exact distributions for the independence and sphericity likelihood ratio test statistics," Journal of Multivariate Analysis, Elsevier, vol. 101(3), pages 583-593, March.
    • Handle: RePEc:eee:jmvana:v:101:y:2010:i:3:p:583-593
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    References listed on IDEAS

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    1. 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.
    2. Coelho, Carlos A., 2004. "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," Journal of Multivariate Analysis, Elsevier, vol. 89(2), pages 191-218, May.
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    Cited by:

    1. Filipe Marques & Carlos Coelho, 2013. "Obtaining the exact and near-exact distributions of the likelihood ratio statistic to test circular symmetry through the use of characteristic functions," Computational Statistics, Springer, vol. 28(5), pages 2091-2115, October.
    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. Filipe J. Marques & Carlos A. Coelho & Paulo C. Rodrigues, 2017. "Testing the equality of several linear regression models," Computational Statistics, Springer, vol. 32(4), pages 1453-1480, December.
    4. Filipe J. Marques & Carlos A. Coelho & Barry C. 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.
    5. Katayama, Shota & Kano, Yutaka & Srivastava, Muni S., 2013. "Asymptotic distributions of some test criteria for the mean vector with fewer observations than the dimension," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 410-421.

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