Testing high-dimensional normality based on classical skewness and Kurtosis with a possible small sample size

By using the idea of principal component analysis, we propose an approach to applying the classical skewness and kurtosis statistics for detecting univariate normality to testing high-dimensional normality. High-dimensional sample data are projected to the principal component directions on which the classical skewness and kurtosis statistics can be constructed. The theory of spherical distributions is employed to derive the null distributions of the combined statistics constructed from the principal component directions. A Monte Carlo study is carried out to demonstrate the performance of the statistics on controlling type I error rates and a simple power comparison with some existing statistics. The effectiveness of the proposed statistics is illustrated by two real-data examples.