This work aimed to determine optimum sample size for the univariate skewness and kurtosis statistics (z1 and z2) adapted to multivariate situation and for the multivariate skewness and kurtosis statistics (k1 and k2) statistics based on simulation. Different probability density functions, univariate and multivariate, were generated by Monte Carlo simulation method to evaluate the type I error rates and the power of the tests. The simulations were done adopting the nominal significance level of 5% and 1%. Situations with p=2, 3, 4 and 5 variables with different correlation structures were evaluated in the case of multivariate distributions. The results showed that k1 statistics is adequate for n> 50, at nominal levels of significance of 5 or 1%; different correlation structured do not affect the power and the type I error rates, the k2 statistics is asymptotically appropriate for kurtosis deviation tests for n> 100, independently of the nominal values of the significance. The skewness statistics, in general, were shown to be more powerful than those of kurtosis, however, the null hypothesis tests of normality must consider both tests jointly, as suggested in the univariate case.
skewness; kurtosis; test for normality; type I error rates and power of the test
