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Limit distributions for Mardia’s measure of multivariate skewness. (English) Zbl 0767.62041
Summary: We study the asymptotic behavior of K. V. Mardia’s [Biometrika 57, 519-530 (1970; Zbl 0214.463), J. R. Stat. Soc., Ser. C 24, 163-171 (1975)] measure of (sample) multivariate skewness. In the special case of an elliptically symmetric distribution, the limit law is a weighted sum of two independent \(\chi^ 2\)-variates. A normal limit distribution arises if the population distribution has positive skewness. These results explain some curiosities in the power performance of a commonly proposed test for multivariate normality based on multivariate skewness.

62H15 Hypothesis testing in multivariate analysis
62H10 Multivariate distribution of statistics
62E20 Asymptotic distribution theory in statistics
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