Shortcomings of Generalized Affine Invariant Skewness Measures

Steffen Gutjahr, Norbert Henze and Martin Folkers

Journal of Multivariate Analysis, 1999, vol. 71, issue 1, 1-23

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.

Keywords: multivariate; skewness; test; for; multivariate; normality; affine; invariance; elliptically; symmetric; distribution (search for similar items in EconPapers)
Date: 1999
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Citations: View citations in EconPapers (6)

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