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

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  • Liang, Jiajuan
  • Tang, Man-Lai
  • Chan, Ping Shing

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.

Suggested Citation

  • Liang, Jiajuan & Tang, Man-Lai & Chan, Ping Shing, 2009. "A generalized Shapiro-Wilk W statistic for testing high-dimensional normality," Computational Statistics & Data Analysis, Elsevier, vol. 53(11), pages 3883-3891, September.
    • Handle: RePEc:eee:csdana:v:53:y:2009:i:11:p:3883-3891
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    References listed on IDEAS

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    1. N. J. H. Small, 1980. "Marginal Skewness and Kurtosis in Testing Multivariate Normality," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 29(1), pages 85-87, March.
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    6. Tan, Ming & Fang, Hong-Bin & Tian, Guo-Liang & Wei, Gang, 2005. "Testing multivariate normality in incomplete data of small sample size," Journal of Multivariate Analysis, Elsevier, vol. 93(1), pages 164-179, March.
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    9. Liang, Jiajuan & Pan, William S.Y. & Yang, Zhen-Hai, 2004. "Characterization-based Q-Q plots for testing multinormality," Statistics & Probability Letters, Elsevier, vol. 70(3), pages 183-190, December.
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