A generalized Shapiro–Wilk WW statistic for testing high-dimensional normality

Shapiro and Wilk’s [Shapiro, S.S., Wilk, M.B., 1965. An analysis of variance test for normality (complete samples). Biometrika 52, 591–611] WW-statistic was found to have competitive power performance in testing univariate normality. Generalizations of the WW-statistic to the multivariate case have been proposed by many researchers. In this paper, we propose a family of generalized WW-statistics for testing high-dimensional normality by using the theory of spherical distributions. The proposed statistics apply to the case that the sample size is smaller than the dimension. Monte Carlo studies demonstrate feasible performance of the proposed tests in controlling type I error rates and power against some non-normal data. It is concluded that the proposed statistics are superior to existing generalizedWW-statistics and show competitive benefits in testing high-dimensional normality with small sample size.