The Mardia measures of multivariate skewness and kurtosis summarize the respective characteristics of a multivariate distribution with two numbers. However, these measures do not reflect the sub-dimensional features of the distribution. Consequently, testing procedures based on these measures may fail to detect skewness or kurtosis present in a sub-dimension of the multivariate distribution. We introduce sub-dimensional Mardia measures of multivariate skewness and kurtosis, and investigate the information they convey about all sub-dimensional distributions of some symmetric and skewed families of multivariate distributions. The maxima of the sub-dimensional Mardia measures of multivariate skewness and kurtosis are considered, as these reflect the maximum skewness and kurtosis present in the distribution, and also allow us to identify the sub-dimension bearing the highest skewness and kurtosis. Asymptotic distributions of the vectors of sub-dimensional Mardia measures of multivariate skewness and kurtosis are derived, based on which testing procedures for the presence of skewness and of deviation from Gaussian kurtosis are developed. The performances of these tests are compared with some existing tests in the literature on simulated and real datasets.
翻译：Mardia多变的二次曲线和二次曲线的测量方法总结了多变的多变分布的特性,但是,这些措施并不反映分布的次维特征,因此,基于这些措施的测试程序可能无法检测到多变分布的次分化中存在的二次曲线或二次曲线。我们引入了多变的次维马迪亚的多变性和二次曲线的测量方法,并调查了它们传递的关于多变分布中某些对称和扭曲的多变分布家庭所有次维分布的信息。考虑了多变的次维马迪亚多变和二次曲线的次维度测量方法的峰值,因为这些方法反映了分布中存在的最大二次曲线和二次曲线的细度,还使我们能够确定与最高二次曲线和二次曲线的二次曲线。关于多变差和二次曲线分布的次维次维分布,它们传递的关于多变差和二次曲线分布的次维度测量方法的分布信息。根据目前测算结果得出了多种变差和二次曲线的亚马迪亚的次维度度测量方法,这些测算数据是根据目前测测测度和模拟的数值测算结果得出的。
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