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

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Abstract

Let X be a d-dimensional random vector having zero expectation and unit covariance matrix. Móri et al. (1993) proposed and studied as a population measure of multivariate skewness. We derive the limit distribution of an affine invariant sample counterpart of . If the distribution of X is spherically symmetric, this limit law is [lambda][chi]d2, where [lambda] depends on EX4 and EX6. In case of spherical (elliptical) symmetry, we also obtain the asymptotic correlation between and Mardia's time-honoured measure of multivariate skewness. If , the limit distribution of is normal. Our results reveal the deficiencies of a test for multivariate normality based on .

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