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On the Non Gaussian Asymptotics of the Likelihood Ratio Test Statistic for Homogeneity of Covariance

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  • Marc Hallin

Abstract

The likelihood ratio test for m-sample homogeneity of covariance is notoriously sensitive to the violations of Gaussian assumptions. Its asymptotic behavior under non-Gaussian densities has been the subject of an abundant literature. In a recent paper, Yanagihara et al. (2005) show that the asymptotic distribution of the likelihood ratio test statistic, under arbitrary elliptical densities with finite fourth-order moments, is that of a linear combination of two mutually independent chi-square variables. Their proof is based on characteristic function methods, and only allows for convergence in distribution conclusions. Moreover, they require homokurticity among the m populations. Exploiting the findings of Hallin and Paindaveine (2008a), we reinforce that convergence-in-distribution result into a convergence-in- probability one —-that is, we explicitly decompose the likelihood ratio test statistic into a linear combination of two variables which are asymptotically independent chi-square —-and moreover extend it to the heterokurtic case.

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Paper provided by ULB -- Universite Libre de Bruxelles in its series Working Papers ECARES with number 2008_039.

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Length: 9 p.
Date of creation: 2008
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Handle: RePEc:eca:wpaper:2008_039

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  1. Nagao, Hisao & Srivastava, M. S., 1992. "On the distributions of some test criteria for a covariance matrix under local alternatives and bootstrap approximations," Journal of Multivariate Analysis, Elsevier, vol. 43(2), pages 331-350, November.
  2. Wakaki, Hirofumi & Eguchi, Shinto & Fujikoshi, Yasunori, 1990. "A class of tests for a general covariance structure," Journal of Multivariate Analysis, Elsevier, vol. 32(2), pages 313-325, February.
  3. Paindaveine, Davy, 2008. "A canonical definition of shape," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2240-2247, October.
  4. Yanagihara, Hirokazu & Tonda, Tetsuji & Matsumoto, Chieko, 2005. "The effects of nonnormality on asymptotic distributions of some likelihood ratio criteria for testing covariance structures under normal assumption," Journal of Multivariate Analysis, Elsevier, vol. 96(2), pages 237-264, October.
  5. Hallin Marc & Paindaveine Davy, 2006. "Parametric and semiparametric inference for shape: the role of the scale functional," Statistics & Risk Modeling, De Gruyter, vol. 24(3), pages 24, December.
  6. Hallin, Marc & Paindaveine, Davy, 2009. "Optimal tests for homogeneity of covariance, scale, and shape," Journal of Multivariate Analysis, Elsevier, vol. 100(3), pages 422-444, March.
  7. Srivastava, M. S. & Khatri, C. G. & Carter, E. M., 1978. "On monotonicity of the modified likelihood ratio test for the equality of two covariances," Journal of Multivariate Analysis, Elsevier, vol. 8(2), pages 262-267, June.
  8. Arjun Gupta & Jin Xu, 2006. "On Some Tests of the Covariance Matrix Under General Conditions," Annals of the Institute of Statistical Mathematics, Springer, vol. 58(1), pages 101-114, March.
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