Abstract
In social and 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 within the stringency framework for skewed alternative space. The stringency concept allows us to rank the tests uniquely. The Bonett and Seier test (Tw) turns out to represent the best statistics for slightly skewed alternatives and the Anderson–Darling (AD); Chen–Shapiro (CS); Shapiro–Wilk (W); and Bispo, Marques, and Pestana (BCMR) statistics are the best choices for moderately skewed alternative distributions. The 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 Tw and Daniele Coin’s COIN-test performed exceptionally well against the highly skewed alternative space.
Highlights
Departures from normality can be measured in a variety of ways; the most common measures are skewness and kurtosis in this regard
For slightly skewed distributions, COIN and W tests exhibit the same power properties (Figures A5 and A6), whereas Wsf and D statistics do not match the standards set by other members of the group (Figures A7 and A8), with maximum power losses of over 50% (Table 2)
The results clearly show that the power loss of these statistics decreases with the increase in (i) sample size and (ii) skewness and kurtosis
Summary
Departures from normality can be measured in a variety of ways; the most common measures are skewness and kurtosis in this regard. Arnau, López-Montiel, Bono, and Bendayan [3] analyzed the shape of 693 real data distributions by including the measures of cognitive ability and other psychological variables in terms of skewness and kurtosis. 5.5% of the distributions were close to the normality assumption Keeping this in mind, the literature has produced few normality tests which are based on skewness and kurtosis [4,5,6,7]. This study is devoted to analyzing the respective impact of change in skewness and kurtosis on the power of normality tests. Symmetry 2019, 11, 872 tests are selected for a comparison of power based on the stringency concept proposed by Islam [16]. Neyman–Pearson (NP) tests are computed against each alternative distribution to construct the power curve. The best test is defined as the test displaying the minimum deviation from the power curve among the maximum deviations of all the tests
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