Abstract
Normality tests are used in the statistical analysis to determine whether a normal distribution is acceptable as a model for the data analysed. A wide range of available tests employs different properties of normal distribution to compare empirical and theoretical distributions. In the present paper, we perform the Monte Carlo simulation to analyse test power. We compare commonly known and applied tests (standard and robust versions of the Jarque-Bera test, Lilliefors test, chi-square goodness-of-fit test, Shapiro-Francia test, Cramer-von Mises goodness-of-fit test, Shapiro-Wilk test, D'Agostino test, and Anderson-Darling test) to the test based on robust L-moments. In the text, in Jarque-Bera type test the moment characteristics of skewness and kurtosis are replaced with their robust versions - L-skewness and L-kurtosis. The distributions with heavy tails (lognormal, Weibull, loglogistic and Student) are used to draw random samples to show the performance of tests when applied on data with outliers. Small sample properties (from 10 observations) are analysed up to large samples of 200 observations. Our results concerning the properties of the classical tests are in line with the conclusion of other recent articles. We concentrate on properties of the test based on L-moments. This normality test is comparable to well-performing and reliable tests; however, it is outperformed by the most powerful Shapiro-Wilks and Shapiro-Francia tests. It works well for Student (symmetric) distribution, comparably with the most frequently used Jarque-Berra tests. As expected, the test is robust to the presence of outliers in comparison with sensitive tests based on product moments or correlations. The test turns out to be very universally reliable.
Highlights
In parametric statistics, the assumption about the probability distribution of data is crucial
We present the relative frequencies of the normality hypothesis rejection for all distributions, including the one that does not meet the requirements for the characteristics used in the test criterion calculation
The lognormal distribution is unimodal and positively skewed with the coefficients of skewness and kurtosis increasing in the parameter σ
Summary
The assumption about the probability distribution of data is crucial. There are numerous statistical methods based on the normal distribution. In addition to exploratory graphs describing the empirical distribution, statistical tests have been designed to determine whether the random sample is drawn from the selected distribution (the normal one in this paper), or if this assumption is at least reasonable. These tests make it possible to assess the suitability of the selected distributions in the form of probabilistic models for the analysed data, not choosing the best of multiple options. Some of them are intended for the normal distribution only (based on its specific properties), others are more generally applicable goodness-of-fit tests
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