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

Abstract

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 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 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.

Keywords:

• 62D05
• 62H10

Keywords:

• Multivariate kurtosis
• multivariate normality test
• normalizing transformation

Acknowledgments

The authors thank the referee for his useful comments. The research of the third author was supported in part by a Grant-in-Aid for Scientific Research (C) (23500360).