A generalized Shapiro-Wilk W statistic for testing high-dimensional normality

Authors:
Jiajuan Liang;Man-Lai Tang;Ping Shing Chan
Affiliations:
Department of Management, University of New Haven, 300 Boston Post Road, West Haven, CT 06516, USA;Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China;Department of Statistics, The Chinese University of Hong Kong, Shatin, New Territory, Hong Kong, China
Venue:
Computational Statistics & Data Analysis
Year:
2009

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Abstract

Shapiro and Wilk's [Shapiro, S.S., Wilk, M.B., 1965. An analysis of variance test for normality (complete samples). Biometrika 52, 591-611] W-statistic was found to have competitive power performance in testing univariate normality. Generalizations of the W-statistic to the multivariate case have been proposed by many researchers. In this paper, we propose a family of generalized W-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 generalizedW-statistics and show competitive benefits in testing high-dimensional normality with small sample size.