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

Abstract

Let X be a d-dimensional random vector having zero expectation and unit covariance matrix. Móri et al. (1993) proposed and studied β ̃ 1,d = |E(|X| 2X)| 2 as a population measure of multivariate skewness. We derive the limit distribution of an affine invariant sample counterpart b ̃ 1,d of β ̃ 1,d . If the distribution of X is spherically symmetric, this limit law is λ χ d 2 , where λ depends on E| X| 4 and E| X| 6. In case of spherical (elliptical) symmetry, we also obtain the asymptotic correlation between b ̃ 1,d and Mardia's time-honoured measure of multivariate skewness. If β ̃ 1,d > 0 , the limit distribution of n 1 2 ( b ̃ 1,d − β ̃ 1,d) is normal. Our results reveal the deficiencies of a test for multivariate normality based on b ̃ 1,d .