Limit distributions for measures of multivariate skewness and kurtosis based on projections

We derive the asymptotic distributions for measures of multivariate skewness and kurtosis defined by Malkovich and Afifi if the underlying distribution is elliptically symmetric. A key step in the derivation is an approximation by suitable Gaussian processes defined on the surface of the unit d-sphere. It is seen that a test for multivariate normality based on skewness in the sense of Malkovich and Afifi is inconsistent against each fixed elliptically symmetric non-normal distribution provided that a weak moment condition holds. Consistency of a test for multinormality based on kurtosis within the class of elliptically symmetric distributions depends on the fourth moment of the marginal distribution of the standardized underlying law. Our results may also be used to give tests for a special elliptically symmetric type against asymmetry or difference in kurtosis.