Limit laws for multivariate skewness in the sense of Móri, Rohatgi and Székely

✍ Scribed by Norbert Henze

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 411 KB
• Volume: 33
• Category: Article
• ISSN: 0167-7152
• DOI: 10.1016/s0167-7152(96)00141-1

✦ Synopsis

Let X be a d-dimensional random vector having zero expectation and unit covariance matrix. Mrri et al. (1993) proposed and studied /~l,a = IIE(IIXII2X)II 2 as a population measure of multivariate skewness. We derive the limit distribution of an affine invariant sample counterpart /~l,a of/~,a. If the distribution of X is spherically symmetric, this limit law is 2X~, where 2 depends on EIIXN 4 and EIIXII 6. In case of spherical (elliptical) symmetry, we also obtain the asymptotic correlation between/~l,a and Mardia's time-honoured measure of multivariate skewness. If ]~l,a > 0, the limit distribution of nl/2(bl,a -/~l,a) is normal. Our results reveal the deficiencies of a test for multivariate normality based on /~l,a.