Comparison of conventional measures of skewness and kurtosis for small sample size

Abstract

The normality assumption can be checked in three ways: graphical methods (histogram, normal Q-Q plot, and boxplots), descriptive statistics (value of skewness and kurtosis) or conducting test of normality (such as Shapiro-Wilk test, Kolmogorow-Smirnow test, Lilliefors test, Jacque-Bera test or Anderson Darling test). This paper focused on the two descriptive statistics which are skewness and kurtosis. A simulation study was carried out to compare the performance for three different types of conventional measures (TYPE 1, TYPE 2, and TYPE 3) of skewness and kurtosis for symmetric and asymmetric distributions. Monte Carlo simulation using R programming language was used to generate data from symmetric and skewed distribution. For symmetric distribution, the performance of TYPE 1, 2 and 3 skewness are comparable. Meanwhile, TYPE 2 kurtosis measure performs better for symmetric normal distribution. For symmetric distribution with negative kurtosis TYPE 1 kurtosis seems to perform better. While for asymmetric distribution, TYPE 2 skewness and kurtosis are better measures. However, all three measures do not perform well for leptokurtic distribution such as t-distribution.