Measures of multivariate skewness and kurtosis in high-dimensional framework

Abstract

Skewness and kurtosis characteristics of a multivariate $p$-dimensional distribution introduced by Mardia (1970) have been used in various testing procedures and demonstrated attractive asymptotic properties in large sample settings. However these characteristics are not designed for high-dimensional problems where the dimensionality, $p$ can largely exceeds the sample size, $N$. Such type of high-dimensional data are commonly encountered in modern statistical applications. This the suggests that new measures of skewness and kurtosis that can accommodate high-dimensional settings must be derived and carefully studied. In this paper, we show that, by exploiting the dependence structure, new expressions for skewness and kurtosis are introduced as an extension of the corresponding Mardia’s measures, which uses the potential advantages that the block-diagonal covariance structure has to offer in high dimensions. Asymptotic properties of newly derived measures are investigated and the cumulant based characterizations are presented along with of applications to a mixture of multivariate normal distributions and multivariate Laplace distribution, for which the explicit expressions of skewness and kurtosis are obtained. Test statistics based on the new measures of skewness and kurtosis are proposed for testing a distribution shape, and their limit distributions are established in the asymptotic framework where $N\to \infty$ and $p$ is fixed but large, including . For the dependence structure learning, the gLasso based technique is explored followed by AIC step which we propose for optimization of the gLasso candidate model. Performance accuracy of the test procedures based on our estimators of skewness and kurtosis are evaluated using Monte Carlo simulations and the validity of the suggested approach is shown for a number of cases when .