Some results on multivariate measures of elliptical and skew-elliptical distributions: higher-order moments, skewness and kurtosis

The kurtosis and skewness of distributions are important measures that can describe the shape of a distribution, and there have been many results for symmetric distributions, but there are still many difficulties and challenges in the characterization of skew distributions. Based on the results of Mardia's and Song's kurtosis measures of elliptical distributions obtained by Zografos [1], we generalize the results and study some measures for elliptical and skew-elliptical distributions. We also derive the expressions of moments of skew-elliptical distributions in terms of the ones of skew-normals and take skew-t, skew-Pearson type VII and skew-Pearson type II distributions as examples.

Automated summary

The kurtosis and skewness are important measures for shape characterization of distributions. While there have been many results for symmetric distributions, characterizing skew distributions still poses difficulties and challenges. Building on previous work on kurtosis measures for elliptical distributions by Zografos [1], we generalize the results and study measures for both elliptical and skew-elliptical distributions. We derive moment expressions for skew-elliptical distributions using those of skew-normal distributions, and provide examples using skew-t, skew-Pearson type VII, and skew-Pearson type II distributions.