Simulation Study on Improved Shapiro-Wilk Tests for Normality

The W statistic proposed by Shapiro and Wilk ([2]) is frequently used for testing of the univariate and multivariate normality. However, the table of coefficients in the W statistic, its critical values and also constants in Johnson’s SB transformation to normal distribution ([3]), are not correct. Royston ([1]) gave an approximation for coefficients in the W statistic and use them to evaluate proper critical values of the Shapiro-Wilk test. In the paper, we determine new constants for the W statistic and Johnson’s SB transformation. Empirical significant levels of the improved Shapiro-Wilk test and the power against chosen alternatives are evaluated via simulation study.