Test for Normality Based on an Edgeworth Expansion

Many statistical procedures require the assumption that the observations in a random sample are drawn from a normal distribution. Several statistical techniques, mostly based on either population moments or empirical distribution functions, are currently available to test whether the observations in a random sample are normally distributed. In this study, we use the Edge-worth series expansion to model deviations from normality, depending on the skewness and excess kurtosis of a normal population. We develop a test statistic based on these two measures to test for normality. However, the sampling distribution of the test statistic is not mathematically tractable. There-fore, we conducted a simulation study by generating the sampling distribution of the proposed test statistic for different sample sizes when the data are normally distributed. The critical values were also calculated at different levels of significance. The power of the proposed test was empirically compared with the Shapiro-Wilk (SW), Shapiro-Francia (SF), Jarque-Bera (JB), Cramer-von Mises (CM), Lilliefors (LF), Kolmogorov-Smirnov (KS), and Anderson-Darling (AD) tests. The proposed test demonstrates competitive power against JB, CM, LF, KS and AD tests for small samples under specific alternatives with low skewness.