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Distribution-free tests of mean vectors and covariance matrices for multivariate paired data


  • Erning Li


  • Johan Lim
  • Kyunga Kim
  • Shin-Jae Lee


We study a permutation procedure to test the equality of mean vectors, homogeneity of covariance matrices, or simultaneous equality of both mean vectors and covariance matrices in multivariate paired data. We propose to use two test statistics for the equality of mean vectors and the homogeneity of covariance matrices, respectively, and combine them to test the simultaneous equality of both mean vectors and covariance matrices. Since the combined test has composite null hypothesis, we control its type I error probability and theoretically prove the asymptotic unbiasedness and consistency of the combined test. The new procedure requires no structural assumption on the covariances. No distributional assumption is imposed on the data, except that the permutation test for mean vector equality assumes symmetric joint distribution of the paired data. We illustrate the good performance of the proposed approach with comparison to competing methods via simulations. We apply the proposed method to testing the symmetry of tooth size in a dental study and to finding differentially expressed gene sets with dependent structures in a microarray study of prostate cancer. Copyright Springer-Verlag 2012

Suggested Citation

  • Erning Li & Johan Lim & Kyunga Kim & Shin-Jae Lee, 2012. "Distribution-free tests of mean vectors and covariance matrices for multivariate paired data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 75(6), pages 833-854, August.
  • Handle: RePEc:spr:metrik:v:75:y:2012:i:6:p:833-854
    DOI: 10.1007/s00184-011-0355-7

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    References listed on IDEAS

    1. Lim, Johan & Li, Erning & Lee, Shin-Jae, 2010. "Likelihood ratio tests of correlated multivariate samples," Journal of Multivariate Analysis, Elsevier, vol. 101(3), pages 541-554, March.
    2. Ming Yuan & Yi Lin, 2007. "Model selection and estimation in the Gaussian graphical model," Biometrika, Biometrika Trust, vol. 94(1), pages 19-35.
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