Definição do tamanho amostral usando simulação Monte Carlos para os testes de normalidade univariado e multivariado baseados em assimetria e curtose

Abstract

An alternative form to verify assumption ofdata normaiity is concerned with the application ofthe tests based on skewness and kurtosis coefficient. The objective of this work was to determine an optimum size samples for the univariate (Zx and Z2) and multivariate (K\ and K2) statistics on basis of simulatíon. The Zi and K\ statistics to verified the skewness alternative Z2 and K2 to verified the kurtosis alternative. Different probability density functions, univariate and multivariate were generated, by Monte Cario simulatíon method with aview to calculating the type I error rates and the power ofthe test. The simulations were done by adopting the nominal levei of5% and 1%. Situations with p=2, 3, 4 and 5 variables, with different correlation structures, were evaluated in the case of multivariates distributions. The evaluation criterion in the univariate case was that ofthe comparison ofthe rates obtained through the value ofthe rates ofempirical power obtained by Shapiro and Wilk test (1965). By Considerering the univariate case, itwas found that the Z\ statistics possesses normal asymptotic approximation for n>25 and a =5% can be recommended for routine use in the univariate case; The Z2 statistics possesses normal asymptotic approximation for n£25 and a =5% can be recommended for routine use in the univariate case for the kurtosis deviation test; The K\ and K2 statistics possess approximation asymptotic better than Zi and Z2 for a lower value ofthe nominal value ofsignificance, recommended for n£25 and n^lOO, respectively, warranting the compromise with the control ofthe type I error rate and elevated power; In the case ofsymmetry distributions wnh close to zero and non-normal the statistics based on skewness deviations present higher power than Shapiro - Wilk's Wstatistics. For the multivariate case the difíerents correlation structures, didnt afíèct me power and type I error rate of the tests; The K\ statistics is adequate for use from n>50 for nominal values of significance of 5 or 1%; The K2 statistics is asymptotically appropriate for kurtosis deviation tests for n>100, independently ofthe nominal values ofthe significance. The sizes indicated in the studies oftype I error rate and power ofthe test, were the same ones found inthe studies ofapproximate percentage points. The conclusion that both in the univariate case and multivariate case the skewness statistics, in general, are more powerful than those ofkurtosis, but the tests ofthe null hypothesis ofnormality must take into account both the tests ofskewness deviations and those ofkurtosis jointly.