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