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Statistica Sinica 12(2002), 769-783



Li-Xing Zhu$^{1,2}$, Kai W. Ng$^1$ and Ping Jing$^3$

$^1$University of Hong Kong, $^2$Chinese Academy of Sciences, Beijing and
$^3$ University of Mining and Technology of China

Abstract: Testing hypotheses on covariance matrices has long been of interest in statistics. The test of homogeneity is very often a preliminary step in discriminant analysis, cluster analysis, MANOVA, etc. In this article we propose non-parametric tests which are based on the eigenvalues of the differences among the sample covariance matrices after a common rescaling. Three resampling techniques for calculating $p$-values are shown to be asymptotically valid: bootstrap, random symmetrization and permutation. Monte Carlo simulations show that the bootstrap performs less satisfactorily than the others in adhering to the nominal level of significance. Some theoretic ground for this phenomenon is given. The simulation results also suggest that the homogeneity tests proposed in this article performs better than the bootstrap version of Bartlett's test.

Key words and phrases: Bartlett homogeneity test, bootstrap, non-parametric tests, permutation test, random symmetrization.

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