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A Jarque-Bera type test for multivariate normality based on second-power skewness and kurtosis
Communications for Statistical Applications and Methods 2021;28:463-475
Published online September 30, 2021
© 2021 Korean Statistical Society.

Namhyun Kim1,a

aDepartment of Science, Hongik University, Korea
Correspondence to: 1 Department of Science, Hongik University, 94 Wausan-Ro, Mapo-Gu, Seoul 04066, Korea. E-mail: nhkim@hongik.ac.kr
Received February 12, 2021; Revised April 7, 2021; Accepted May 17, 2021.
Desgagné and de Micheaux (2018) proposed an alternative univariate normality test to the Jarque-Bera test. The proposed statistic is based on the sample second power skewness and kurtosis while the Jarque-Bera statistic uses sample Pearson’s skewness and kurtosis that are the third and fourth standardized sample moments, respectively. In this paper, we generalize their statistic to a multivariate version based on orthogonalization or an empirical standardization of data. The proposed multivariate statistic follows chi-squared distribution approximately. A simulation study shows that the proposed statistic has good control of type I error even for a very small sample size when critical values from the approximate distribution are used. It has comparable power to the multivariate version of the Jarque-Bera test with exactly the same idea of the orthogonalization. It also shows much better power for some mixed normal alternatives.
Keywords : Goodness of fit test, Jarque-Bera test, second power kurtosis, second power skewness, multivariate normality, power comparison