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Testing the hypothesis of a block compound symmetric covariance matrix for elliptically contoured distributions

Author

Listed:
  • Carlos A. Coelho

    (Universidade Nova de Lisboa)

  • Anuradha Roy

    (The University of Texas at San Antonio)

Abstract

In this paper, the authors study the problem of testing the hypothesis of a block compound symmetry covariance matrix with two-level multivariate observations, taken for m variables over u sites or time points. Through the use of a suitable block-diagonalization of the hypothesis matrix, it is possible to obtain a decomposition of the main hypothesis into two sub-hypotheses. Using this decomposition, it is then possible to obtain the likelihood ratio test statistic as well as its exact moments in a much simpler way. The exact distribution of the likelihood ratio test statistic is then analyzed. Because this distribution is quite elaborate, yielding a non-manageable distribution function, a manageable but very precise near-exact distribution is developed. Numerical studies conducted to evaluate the closeness between this near-exact distribution and the exact distribution show the very good performance of this approximation even for very small sample sizes and the approach followed allows us to extend its validity to situations where the population distributions are elliptically contoured. A real-data example is presented and a simulation study is also conducted.

Suggested Citation

  • Carlos A. Coelho & Anuradha Roy, 2017. "Testing the hypothesis of a block compound symmetric covariance matrix for elliptically contoured distributions," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(2), pages 308-330, June.
    • Handle: RePEc:spr:testjl:v:26:y:2017:i:2:d:10.1007_s11749-016-0512-4
    DOI: 10.1007/s11749-016-0512-4
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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. Arnold, Barry C. & Coelho, Carlos A. & Marques, Filipe J., 2013. "The distribution of the product of powers of independent uniform random variables — A simple but useful tool to address and better understand the structure of some distributions," Journal of Multivariate Analysis, Elsevier, vol. 113(C), pages 19-36.
    3. Dale Zimmerman & Vicente Núñez-Antón & Timothy Gregoire & Oliver Schabenberger & Jeffrey Hart & Michael Kenward & Geert Molenberghs & Geert Verbeke & Mohsen Pourahmadi & Philippe Vieu & Dela Zimmerman, 2001. "Parametric modelling of growth curve data: An overview," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 10(1), pages 1-73, June.
    4. 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.
    5. 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.
    6. Larissa A. Matos & Luis M. Castro & Víctor H. Lachos, 2016. "Censored mixed-effects models for irregularly observed repeated measures with applications to HIV viral loads," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(4), pages 627-653, December.
    7. 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.
    8. Annie Qu & Runze Li, 2006. "Quadratic Inference Functions for Varying-Coefficient Models with Longitudinal Data," Biometrics, The International Biometric Society, vol. 62(2), pages 379-391, June.
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