Evaluation of Techniques for Univariate Normality Test Using Monte Carlo Simulation

摘要

This paper examines the sensitivity of nine normality test statistics; W/S, Jaque-Bera, Adjusted Jaque-Bera, D'Agostino, Shapiro-Wilk, Shapiro-Francia, Ryan-Joiner, Lilliefors' and Anderson Darlings test statistics, with a view to determining the effectiveness of the techniques to accurately determine whether a set of data is from normal distribution or not. Simulated data of sizes 5, 10, ..., 100 is used for the study and each test is repeated 100 times for increased reliability. Data from normal distributions (N (2, 1) and N (0, 1)) and non-normal distributions (asymmetric and symmetric distributions: Weibull, Chi-Square, Cauchy and t-distributions) are simulated and tested for normality using the nine normality test statistics. To ensure uniformity of results, one statistical software is used in all the data computations to eliminate variations due to statistical software. The error rate of each of the test statistic is computed; the error rate for the normal distribution is the type I error and that for non-normal distribution is type II error. Power of test is computed for the non-normal distributions and use to determine the strength of the methods. The ranking of the nine normality test statistics in order of superiority for small sample sizes is; Adjusted Jarque-Bera, Lilliefor's, D'Agostino, Ryan-Joiner, Shapiro-Francia, Shapiro-Wilk, W/S, Jarque-Bera and Anderson-Darling test statistics while for large sample sizes, we have; D'Agostino, Ryan-Joiner, Shapiro-Francia, Jarque-Bera, Anderson-Darling, Lilliefor's, Adjusted Jarque-Bera, Shapiro-Wilk and W/S test statistics. Hence, only D'Agostino test statistic is classified as Uniformly Most Powerful since it is effective for both small and large sample sizes. Other methods are Locally Most Powerful. Shapiro-Francia, an improvement of Shapiro-Wilk is more sensitive for both small and large samples, hence should replace Shapiro-Wilk while the Adjsted Jarque-Bera and the Jarque-Bera should both be retained for small and large samples respectively.