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Testing Multivariate Normality Based on Beta-Representative Points

Author

Listed:
  • Yiwen Cao

    (Department of Statistics and Data Science, BNU-HKBU United International College, Zhuhai 519087, China)

  • Jiajuan Liang

    (Department of Statistics and Data Science, BNU-HKBU United International College, Zhuhai 519087, China
    Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science, BNU-HKBU United International College, Zhuhai 519087, China)

  • Longhao Xu

    (Department of Medical Statistics, University Medical Center Göettingen, 37075 Göettingen, Germany)

  • Jiangrui Kang

    (Department of Statistics and Data Science, BNU-HKBU United International College, Zhuhai 519087, China)

Abstract

Testing multivariate normality in high-dimensional data analysis has been a long-lasting topic in the area of goodness of fit. Numerous methods for this purpose can be found in the literature. Reviews on different methods given by influential researchers show that new methods keep emerging in the literature from different perspectives. The theory of statistical representative points provides a new perspective to construct tests for multivariate normality. To avoid the difficulty and huge computational load in finding the statistical representative points from a high-dimensional probability distribution, we develop an approach to constructing a test for high-dimensional normal distribution based on the representative points of the simple univariate beta distribution. The representative-points-based approach is extended to the the case that the sample size may be smaller than the dimension. A Monte Carlo study shows that the new test is able to control type I error rates fairly well for both large and small sample sizes when faced with a high dimension. The power of the new test against some non-normal distributions is generally or substantially improved for a set of selected alternative distributions. A real-data example is given for a simple application illustration.

Suggested Citation

  • Yiwen Cao & Jiajuan Liang & Longhao Xu & Jiangrui Kang, 2024. "Testing Multivariate Normality Based on Beta-Representative Points," Mathematics, MDPI, vol. 12(11), pages 1-16, May.
  • Handle: RePEc:gam:jmathe:v:12:y:2024:i:11:p:1711-:d:1405870
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    References listed on IDEAS

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    1. Norbert Henze, 2002. "Invariant tests for multivariate normality: a critical review," Statistical Papers, Springer, vol. 43(4), pages 467-506, October.
    2. Bruno Ebner & Norbert Henze, 2020. "Tests for multivariate normality—a critical review with emphasis on weighted $$L^2$$ L 2 -statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(4), pages 845-892, December.
    3. Romeu, J. L. & Ozturk, A., 1993. "A Comparative Study of Goodness-of-Fit Tests for Multivariate Normality," Journal of Multivariate Analysis, Elsevier, vol. 46(2), pages 309-334, August.
    4. Bruno Ebner & Norbert Henze, 2020. "Rejoinder on: Tests for multivariate normality—a critical review with emphasis on weighted $$L^2$$ L 2 -statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(4), pages 911-913, December.
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