An analysis of multivariate measures of skewness and kurtosis of skew-elliptical distributions

TL;DR

This paper derives explicit formulas for various skewness and kurtosis measures of skew-elliptical distributions, compares them through simulations, and demonstrates their application on real data.

Contribution

It provides exact expressions for multiple skewness and kurtosis measures for skew-elliptical distributions, enhancing their analysis and application.

Findings

Derived formulas for skewness and kurtosis measures.

Compared measures through Monte Carlo simulations.

Applied measures to real data for validation.

Abstract

This paper examines eight measures of skewness and Mardia measure of kurtosis for skew-elliptical distributions. Multivariate measures of skewness considered include Mardia, Malkovich-Afifi, Isogai, Song, Balakrishnan-Brito-Quiroz, M$\acute{o}$ri, Rohatgi and Sz$\acute{e}$kely, 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 $t$, 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 Monte Carlo 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.