Ranking of Normality Tests—An Appraisal through Skewed Alternative Space

In social & health sciences, many statistical procedures and estimation techniques rely on the underlying distributional assumption of normality of the data. Non-normality may lead to incorrect statistical inferences. This study evaluates the performance of selected normality tests on the stringency framework for the skewed alternative space. Stringency concept allows us to rank the tests uniquely. Bonett & Seier test (T_w) turns out to be the best statistics for slightly skewed alternatives and the Anderson-Darling (AD), Chen-Shapiro (CS), Shapiro-Wilk (W) and Bispo, Marques, & Pestana, (BCMR) statistics are the best choices for moderately skewed alternative distributions. Maximum loss of Jarque-Bera (JB) and its robust form (RJB), in terms of deviations from the power envelope, is greater than 50% even for large sample sizes which makes them less attractive in testing the hypothesis of normality against the moderately skewed alternatives. On balance, all selected normality tests except T_w and COIN performed exceptionally well against the highly skewed alternative space.