Shortcomings of Generalized Affine Invariant Skewness Measures

✍ Scribed by Steffen Gutjahr; Norbert Henze; Martin Folkers

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 174 KB
• Volume: 71
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1999.1823

✦ Synopsis

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 / 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.