An alternative form to verify assumption of data normality is concerned with the application of the tests based on skewness and kurtosis coefficients. The objective of this work was to determine an optimum sample size for the univariate (Z1 and Z2) and multivariate (K1 and K2) statistics on basis of simulation. The Z1 and Z2 statistics are related to the skewness and the Z1 and Z2 are related to the kurtosis. Different univariate probability density functions were generated, by Monte Carlo simulation method with a view to calculating the type I error rates and the power of the test. The simulations were done by adopting the probability level of 5% and 1%. The evaluation criterion in the univariate case was that of the comparison of the rates obtained through the value of the rates of empirical power obtained by Shapiro & Wilk (1965) test. By considering the univariate case, it was found that the Z1e Z2 statistics possess normal asymptotic approximation for n>25 and α=5% can be recommended for routine use in the univariate case. The K1 and K2 statistics possess approximation asymptotic better than Z1 and Z2 for a lower value of the nominal value of significance, recommended for n>25 and n>100, respectively, warranting the compromise with the control of the type I error rate and elevated power. In the case of symmetry distributions with efficient of skewness close to zero and non-normal, the statistics based on skewness deviations present higher power than Shapiro - Wilk's W statistics. It is concluded that the skewness statistic in general, is more powerful than that of kurtosis, but the tests of the null hypothesis of normality must take into account both the tests of skewness deviations and those of kurtosis jointly.
Skewness; kurtosis; test for normality; type I error rates and power of the test
