Multivariate measures of skewness and kurtosis for skew-elliptical distributions

This article examines eight measures of skewness and the Mardia measure of kurtosis for skew-elliptical distributions. Multivariate measures of skewness considered include Mardia, Malkovich-Afifi, Isogai, Song, Balakrishnan-Brito-Quiroz, Móri-Rohatgi-Székely, Kollo, and Srivastava measures. We first study the canonical form of skew-elliptical distributions and then derive exact expressions of all measures of skewness and kurtosis for the family of skew-elliptical distributions, except for Song’s measure. Specifically, the formulas of these measures for skew normal, skew <i>t</i>, skew logistic, skew Laplace, skew Pearson type II, and skew Pearson type VII distributions are obtained. Next, as in Malkovich and Afifi (1973), test statistics based on a random sample are constructed for illustrating the usefulness of the established results. In a simulation study, different measures of skewness and kurtosis for 2-dimensional skewed distributions are calculated and compared. Finally, real data is analyzed to demonstrate all the results.