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.

翻译：本文研究了偏斜椭圆分布的八种偏度度量及Mardia峰度度量。所考虑的多元偏度度量包括Mardia、Malkovich-Afifi、Isogai、Song、Balakrishnan-Brito-Quiroz、Móri、Rohatgi-Székely、Kollo和Srivastava度量。我们首先研究偏斜椭圆分布的典型形式，然后推导出除Song度量外，所有偏度和峰度度量在偏斜椭圆分布族中的精确表达式。具体地，获得了偏正态分布、偏t分布、偏logistic分布、偏Laplace分布、偏Pearson II型分布和偏Pearson VII型分布的这些度量公式。接着，如同Malkovich和Afifi（1973）的方法，基于随机样本构建检验统计量以说明所建立结果的有效性。通过蒙特卡洛模拟研究，计算并比较了二维偏斜分布的不同偏度和峰度度量。最后，通过真实数据分析展示所有结果。