Shortcomings of generalized affine invariant skewness measures

JOURNAL OF MULTIVARIATE ANALYSIS

卷:71

Article

Gutjahr, S ; Henze, N ; Folkers, M

关键词: multivariate skewness;    test for multivariate normality;    affine invariance;    elliptically symmetric distribution;   

DOI  :  10.1006/jmva.1999.1823

来源: Elsevier

【 摘 要 】

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. (C) 1999 Academic Press.

【 授权许可】

Free