Goodness-of-fit testing for normality where alternative distributions have undefined or constant skewness and excess kurtosis

Abstract

The aim of the paper is to examine goodness-of-fit testing for normality where alternative distributions have undefined or constant skewness and excess kurtosis. The first step involves collecting a set of normality-oriented goodness-of-fit tests (GoFTs) recommended for use in the source literature. The second step is to create a family of distributions with undefined or constant skewness and excess kurtosis. For greater clarity, the alternative distributions have been divided into three groups. Group I consisted of symmetric alternatives with undefined excess kurtosis, group II of symmetric alternatives with constant non-zero excess kurtosis, and group III comprised asymmetric alternatives with constants non-zero skewness and non-zero excess kurtosis. An appropriate similarity measure (SM) of a given alternative distribution to the normal distribution was applied. The third step involved a Monte Carlo simulation. The obtained results indicate that the GoFT power for the analysed alternatives and for a given sample size remain relatively unchanged despite the significant decrease in SM. Several GoFTs proved incapable of making a distinction between normal distribution and the alternatives. This situation occurs even when SM equals 0.15. If SM equals 1, then the probability density functions (PDFs) are identical. The best tests for group I are the Gel-Miao-Gastwirth (SJ) and Robust Jarque- Bera (RJB) tests (positive excess kurtosis) as well as the Coin and Chen-Shapiro (CS) tests (negative excess kurtosis), for group II the SJ, 1st Hosking and RJB tests, while the Zhang-Wu (ZA) and CS tests are best for group III.