On the distributions of multivariate sample skewness

In this paper, we consider the multivariate normality test based on measure of multivariate sample skewness defined by Srivastava (1984). Srivastava derived asymptotic expectation up to the order N−1 for the multivariate sample skewness and approximate χ2χ2 test statistic, where N   is sample size. Under normality, we derive another expectation and variance for Srivastava's multivariate sample skewness in order to obtain a better test statistic. From this result, improved approximate χ2χ2 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 obtained expectation, variance and improved approximate χ2χ2 test statistic. Furthermore, upper and lower percentiles of χ2χ2 test statistic derived in this paper are compared with those of χ2χ2 test statistic derived by Mardia (1974) which is used multivariate sample skewness defined by Mardia (1970).