Journal of Statistical Theory and Practice

, Volume 8, Issue 2, pp 382–399 | Cite as

Modified Jarque-Bera Type Tests for Multivariate Normality in a High-Dimensional Framework

  • Kazuyuki KoizumiEmail author
  • Masashi Hyodo
  • Tatjana Pavlenko


In this article, we introduce two types of new omnibus procedures for testing multivariate normality based on the sample measures of multivariate skewness and kurtosis. These characteristics, initially introduced by, for example, Mardia (1970) and Srivastava (1984), were then extended by Koizumi, Okamoto, and Seo (2009), who proposed the multivariate Jarque-Bera type test (MJB1) based on the Srivastava (1984) principal components measure scores of skewness and kurtosis. We suggest an improved MJB test (MJB2) that is based on the Wilson-Hilferty transform, and a modified MJB test (mMJB) that is based on the F-approximation to mMJB. Asymptotic properties of both tests are examined, assuming that both dimensionality and sample size go to infinity at the same rate. Our simulation study shows that the suggested mMJB test outperforms both MJB1 and MJB2 for a number of high-dimensional scenarios. The mMJB test is then used for testing multivariate normality of the real data digitalized character image.


Jarque-Bera test Multivariate skewness Multivariate kurtosis Normality test Normalizing transformation 

AMS Subject Classification: Primary


AMS Subject Classification: Secondary



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Enomoto, R. 2010. On the distribution of improved multivariate Jarque-Bera test statistics. Master’s thesis, Tokyo University of Science, Tokyo, Japan.Google Scholar
  2. Hastie, T., R. Tibshirani, and J. Friedman. 2009. The elements of statistical learning. Springer.Google Scholar
  3. Jarque, C. M., and A. K. Bera. 1987. A test for normality of observations and regression residuals. Int. Stat. Rev., 55, 163–172.MathSciNetCrossRefGoogle Scholar
  4. Klar, B. 2002. A treatment of multivariate skewness, kurtosis, and related statistics. J. Multivariate Anal., 83, 141–165.MathSciNetCrossRefGoogle Scholar
  5. Koizumi, K., N. Okamoto, and T. Seo. 2009. On Jarque-Bera tests for assessing multivariate normality. J. Stat. Adv. Theory Appl., 1, 207–220.zbMATHGoogle Scholar
  6. Kollo, T. 2008. Multivariate skewness and kurtosis measures with an application in ICA. J. Multivariate Anal., 99, 2328–2338.MathSciNetCrossRefGoogle Scholar
  7. Malkovich, J. R., and A. A. Afifi. 1973. On tests for multivariate normality. J. Am. Stat. Assoc., 68, 176–179.CrossRefGoogle Scholar
  8. Mardia, K. V. 1970. Measures of multivariate skewness and kurtosis with applications. Biometrika, 57, 519–530.MathSciNetCrossRefGoogle Scholar
  9. Mardia, K. V., and K. Foster. 1983. Omnibus tests of multinormality based on skewness and kurtosis. Commun. Stat. Theory Methods, 12, 207–221.MathSciNetCrossRefGoogle Scholar
  10. Mudholkar, G. S., M. McDermott, and D. K. Srivastava. 1992. A test of p-variate normality. Biometrika, 79, 850–854.MathSciNetCrossRefGoogle Scholar
  11. Poitras, G. 2006. More on the correct use of omnibus tests for normality. Econ. Lett., 90, 304–309.MathSciNetCrossRefGoogle Scholar
  12. Royston, J. P. 1983. Some techniques for assessing multivariate normality based on the Shapiro-Wilk W. Appl. Stat., 32, 121–133.CrossRefGoogle Scholar
  13. Seo, T., and M. Ariga. 2011. On the distribution of sample measure of multivariate kurtosis. J. Combinatorics Information System Sci., 36, 179–200.zbMATHGoogle Scholar
  14. Shapiro, S. S., and M. B. Wilk. 1965. An analysis of variance test for normality (complete samples). Biometrika, 52, 591–611.MathSciNetCrossRefGoogle Scholar
  15. Srivastava, M. S. 1984. A measure of skewness and kurtosis and a graphical method for assessing multivariate normality. Stat. Prob. Lett., 2, 263–267.MathSciNetCrossRefGoogle Scholar
  16. Srivastava, M. S., and T. K. Hui. 1987. On assessing multivariate normality based on Shapiro-Wilk W statistic. Stat. Prob. Lett., 5, 15–18.MathSciNetCrossRefGoogle Scholar
  17. Urzúa, C. M. 1996. On the correct use of omnibus tests for normality. Econ. Lett., 90, 304–309.MathSciNetzbMATHGoogle Scholar
  18. Wilson, E. B., and M. M. Hilferty. 1931. The distribution of chi-square. Mathematics, 17, 684–688.zbMATHGoogle Scholar

Copyright information

© Grace Scientific Publishing 2014

Authors and Affiliations

  • Kazuyuki Koizumi
    • 1
    Email author
  • Masashi Hyodo
    • 2
  • Tatjana Pavlenko
    • 3
  1. 1.International College of Arts and SciencesYokohama City UniversityYokohama City, KanagawaJapan
  2. 2.Graduate School of EconomicsUniversity of TokyoTokyoJapan
  3. 3.Department of MathematicsKTH Royal Institute of TechnologyStockholmSweden

Personalised recommendations