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Shortcomings of Generalized Affine Invariant Skewness Measures

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  • Gutjahr, Steffen
  • Henze, Norbert
  • Folkers, Martin

Abstract

This paper studies the asymptotic behavior of a generalization of Mardia's affine invariant measure of (sample) multivariate skewness. If the underlying distribution is elliptically symmetric, the limiting distribution is a finite sum of weighted independent [chi]2-variates, and the weights are determined by three moments of the radial distribution of the corresponding spherically symmetric generator. If the population distribution has positive generalized skewness a normal limiting distribution occurs. The results clarify the shortcomings of generalized skewness measures when used as statistics for testing for multivariate normality. Loosely speaking, normality will be falsely accepted for a short-tailed non-normal elliptically symmetric distribution, and it will be correctly rejected for a long-tailed non-normal elliptically symmetric distribution. The wrong diagnosis in the latter case, however, would be rejection due to positive skewness.

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  • Gutjahr, Steffen & Henze, Norbert & Folkers, Martin, 1999. "Shortcomings of Generalized Affine Invariant Skewness Measures," Journal of Multivariate Analysis, Elsevier, vol. 71(1), pages 1-23, October.
    • Handle: RePEc:eee:jmvana:v:71:y:1999:i:1:p:1-23
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    References listed on IDEAS

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    1. Norbert Henze, 1997. "Do components of smooth tests of fit have diagnostic properties?," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 45(1), pages 121-130, January.
    2. L. Baringhaus & N. Henze, 1988. "A consistent test for multivariate normality based on the empirical characteristic function," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 35(1), pages 339-348, December.
    3. Henze, Norbert, 1997. "Limit laws for multivariate skewness in the sense of Móri, Rohatgi and Székely," Statistics & Probability Letters, Elsevier, vol. 33(3), pages 299-307, May.
    4. Henze, Norbert & Wagner, Thorsten, 1997. "A New Approach to the BHEP Tests for Multivariate Normality," Journal of Multivariate Analysis, Elsevier, vol. 62(1), pages 1-23, July.
    5. Bera, A. & John, S., 1983. "Tests for multivariate normality with Pearson alternatives," LIDAM Reprints CORE 534, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

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    2. Alexander, Carol & Meng, Xiaochun & Wei, Wei, 2022. "Targeting Kollo skewness with random orthogonal matrix simulation," European Journal of Operational Research, Elsevier, vol. 299(1), pages 362-376.
    3. Klar, Bernhard, 2002. "A Treatment of Multivariate Skewness, Kurtosis, and Related Statistics," Journal of Multivariate Analysis, Elsevier, vol. 83(1), pages 141-165, October.
    4. Hanke, Michael & Penev, Spiridon & Schief, Wolfgang & Weissensteiner, Alex, 2017. "Random orthogonal matrix simulation with exact means, covariances, and multivariate skewness," European Journal of Operational Research, Elsevier, vol. 263(2), pages 510-523.
    5. Norbert Henze, 2002. "Invariant tests for multivariate normality: a critical review," Statistical Papers, Springer, vol. 43(4), pages 467-506, October.
    6. Carol Alexander & Xiaochun Meng & Wei Wei, 2020. "Targetting Kollo Skewness with Random Orthogonal Matrix Simulation," Papers 2004.06586, arXiv.org, revised Sep 2021.

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