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Estimation of multivariate 3rd moment for high-dimensional data and its application for testing multivariate normality

Author

Listed:
  • Takayuki Yamada

    (Kagoshima University)

  • Tetsuto Himeno

    (Shiga University)

Abstract

This paper is concerned with the multivariate 3rd moment and its estimation. Mardia (Biometrika 57:519–530, 1970) and Srivastava (Stat Probab Lett 2:263–267, 1984) proposed the multivariate skewness and its estimator, independently. However, these estimators cannot be defined for the case in which the dimension p is larger than the sample size N. In this paper, we treat the multivariate 3rd moment $$\gamma $$ γ which is defined by using Hadamard product of observation vectors, and propose an estimate of $$\gamma $$ γ which is well defined when $$p>N$$ p > N . Based on the estimator, we propose a new test for multivariate normality. Under the null hypothesis, the test statistic is asymptotically standard normal, which is supported by Monte Carlo simulations. We calculate some empirical powers to see the performance of the test.

Suggested Citation

  • Takayuki Yamada & Tetsuto Himeno, 2019. "Estimation of multivariate 3rd moment for high-dimensional data and its application for testing multivariate normality," Computational Statistics, Springer, vol. 34(2), pages 911-941, June.
    • Handle: RePEc:spr:compst:v:34:y:2019:i:2:d:10.1007_s00180-018-00865-9
    DOI: 10.1007/s00180-018-00865-9
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    References listed on IDEAS

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    1. Kubokawa, Tatsuya & Srivastava, Muni S., 2008. "Estimation of the precision matrix of a singular Wishart distribution and its application in high-dimensional data," Journal of Multivariate Analysis, Elsevier, vol. 99(9), pages 1906-1928, October.
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