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Tests for multivariate analysis of variance in high dimension under non-normality

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  • Srivastava, Muni S.
  • Kubokawa, Tatsuya

Abstract

In this article, we consider the problem of testing the equality of mean vectors of dimension p of several groups with a common unknown non-singular covariance matrix Σ, based on N independent observation vectors where N may be less than the dimension p. This problem, known in the literature as the multivariate analysis of variance (MANOVA) in high-dimension has recently been considered in the statistical literature by Srivastava and Fujikoshi (2006) [8], Srivastava (2007) [5] and Schott (2007) [3]. All these tests are not invariant under the change of units of measurements. On the lines of Srivastava and Du (2008) [7] and Srivastava (2009) [6], we propose a test that has the above invariance property. The null and the non-null distributions are derived under the assumption that (N,p)→∞ and N may be less than p and the observation vectors follow a general non-normal model.

Suggested Citation

  • Srivastava, Muni S. & Kubokawa, Tatsuya, 2013. "Tests for multivariate analysis of variance in high dimension under non-normality," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 204-216.
    • Handle: RePEc:eee:jmvana:v:115:y:2013:i:c:p:204-216
    DOI: 10.1016/j.jmva.2012.10.011
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    References listed on IDEAS

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    1. Srivastava, Muni S., 2009. "A test for the mean vector with fewer observations than the dimension under non-normality," Journal of Multivariate Analysis, Elsevier, vol. 100(3), pages 518-532, March.
    2. Schott, James R., 2007. "Some high-dimensional tests for a one-way MANOVA," Journal of Multivariate Analysis, Elsevier, vol. 98(9), pages 1825-1839, October.
    3. Srivastava, Muni S. & Fujikoshi, Yasunori, 2006. "Multivariate analysis of variance with fewer observations than the dimension," Journal of Multivariate Analysis, Elsevier, vol. 97(9), pages 1927-1940, October.
    4. Srivastava, Muni S. & Du, Meng, 2008. "A test for the mean vector with fewer observations than the dimension," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 386-402, March.
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    Cited by:

    1. Zhang, Jin-Ting & Zhou, Bu & Guo, Jia, 2022. "Linear hypothesis testing in high-dimensional heteroscedastic one-way MANOVA: A normal reference L2-norm based test," Journal of Multivariate Analysis, Elsevier, vol. 187(C).
    2. Jamshid Namdari & Debashis Paul & Lili Wang, 2021. "High-Dimensional Linear Models: A Random Matrix Perspective," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(2), pages 645-695, August.
    3. Zhang, Jin-Ting & Guo, Jia & Zhou, Bu, 2017. "Linear hypothesis testing in high-dimensional one-way MANOVA," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 200-216.
    4. Friedrich, Sarah & Pauly, Markus, 2018. "MATS: Inference for potentially singular and heteroscedastic MANOVA," Journal of Multivariate Analysis, Elsevier, vol. 165(C), pages 166-179.
    5. Jiang Hu & Zhidong Bai & Chen Wang & Wei Wang, 2017. "On testing the equality of high dimensional mean vectors with unequal covariance matrices," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(2), pages 365-387, April.
    6. Muni S. Srivastava & Hirokazu Yanagihara & Tatsuya Kubokawa, 2014. "Tests for Covariance Matrices in High Dimension with Less Sample Size," CIRJE F-Series CIRJE-F-933, CIRJE, Faculty of Economics, University of Tokyo.
    7. Tianming Zhu & Jin-Ting Zhang, 2022. "Linear hypothesis testing in high-dimensional one-way MANOVA: a new normal reference approach," Computational Statistics, Springer, vol. 37(1), pages 1-27, March.
    8. Watanabe, Hiroki & Hyodo, Masashi & Nakagawa, Shigekazu, 2020. "Two-way MANOVA with unequal cell sizes and unequal cell covariance matrices in high-dimensional settings," Journal of Multivariate Analysis, Elsevier, vol. 179(C).
    9. Harrar, Solomon W. & Kong, Xiaoli, 2016. "High-dimensional multivariate repeated measures analysis with unequal covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 145(C), pages 1-21.
    10. Zhang, Jin-Ting & Zhu, Tianming, 2022. "A new normal reference test for linear hypothesis testing in high-dimensional heteroscedastic one-way MANOVA," Computational Statistics & Data Analysis, Elsevier, vol. 168(C).
    11. Huiqin Li & Jiang Hu & Zhidong Bai & Yanqing Yin & Kexin Zou, 2017. "Test on the linear combinations of mean vectors in high-dimensional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(1), pages 188-208, March.
    12. Zimmermann, Georg & Pauly, Markus & Bathke, Arne C., 2020. "Multivariate analysis of covariance with potentially singular covariance matrices and non-normal responses," Journal of Multivariate Analysis, Elsevier, vol. 177(C).
    13. Ley, Christophe & Paindaveine, Davy & Verdebout, Thomas, 2015. "High-dimensional tests for spherical location and spiked covariance," Journal of Multivariate Analysis, Elsevier, vol. 139(C), pages 79-91.
    14. Harrar, Solomon W. & Kong, Xiaoli, 2022. "Recent developments in high-dimensional inference for multivariate data: Parametric, semiparametric and nonparametric approaches," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    15. Jin-Ting Zhang & Bu Zhou & Jia Guo, 2022. "Testing high-dimensional mean vector with applications," Statistical Papers, Springer, vol. 63(4), pages 1105-1137, August.
    16. Davy Paindaveine & Thomas Verdebout, 2013. "Universal Asymptotics for High-Dimensional Sign Tests," Working Papers ECARES ECARES 2013-40, ULB -- Universite Libre de Bruxelles.
    17. Ping‐Shou Zhong & Jun Li & Piotr Kokoszka, 2021. "Multivariate analysis of variance and change points estimation for high‐dimensional longitudinal data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(2), pages 375-405, June.

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