Journal of Statistical Theory and Practice

, Volume 7, Issue 1, pp 107–119 | Cite as

On the Asymptotic Normality of Test Statistics Using Song’s Kurtosis

  • Rie EnomotoEmail author
  • Naoya Okamoto
  • Takashi Seo


In this article, we consider the multivariate normality test based on the sample measure of multivariate kurtosis defined by Song (2001). We calculate moments of Song’s kurtosis when covariance matrix Σ is known and unknown. In addition, we derive expectations and variances of the multivariate sample kurtosis under normality and propose three standardized test statistics using these expectations and variances. When Σ is known, we also propose a test statistic whose approximate accuracy for multivariate normal distribution is improved rather than the standardized statistics by normalizing transformation. In order to evaluate accuracy of proposed test statistics, the numerical results by Monte Carlo simulation for some selected values of parameters are presented.


Multivariate kurtosis multivariate normality test normalizing transformation 

AMS Subject Classifications

62D05 62H10 


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Copyright information

© Grace Scientific Publishing 2013

Authors and Affiliations

  1. 1.Department of Mathematical Information Science, Graduate School of ScienceTokyo University of ScienceTokyoJapan
  2. 2.Department of Food Sciences, Faculty of Health and NutritionTokyo Seiei CollegeTokyoJapan

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