On the Distribution of Multivariate Sample Skewness for Assessing Multivariate Normality

In this paper, we consider the multivariate normality test based on measure of multivariate sample skewness defined by Srivastava (1984). Srivastava derived asymp-totic expectation up to the order N−1 for the multivariate sample skewness and ap-proximate χ2 test statistic, where N is sample size. Under normality, we derive the exact expectation and variance for Srivastava’s multivariate sample skewness. From this result, improved approximate χ2 test statistic using the multivariate sample skewness is also given for assessing multivariate normality. Finally, the numerical result by Monte Carlo simulation is shown in order to evaluate accuracy of the ob-tained expectation, variance and improved approximate χ2 test statistic. Key Words: asymptotic distribution; multivariate normal distribution; approximate χ2 test statistic