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

Abstract

The likelihood ratio test for m-sample homogeneity of covariance is notoriously sensitive to violations of the 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 (2009), 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 that are asymptotically independent chi-square– and moreover extend it to the heterokurtic case.