  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.
