Asian Journal of Probability and Statistics

Assessing the normality of regression residuals is an important component of regression diagnostics because departures from normality can affect statistical inference, particularly in small and moderate samples. This study compared the empirical Type I error rates and empirical power of six selected normality tests for regression residuals using a Monte Carlo simulation framework with 1,000 replications. Data were generated from a simple linear regression model for sample sizes of 10, 20, 30, 40, 50, and 100 under six residual distributions: normal, exponential, Student’s t, Cauchy, bimodal, and contaminated normal. The Energy, Anderson–Darling, Cramér–von Mises, Lilliefors, Shapiro–Francia, and Random Projection tests were evaluated at a nominal significance level of 5%. Under normal residuals, all six procedures produced empirical Type I error rates close to the nominal level. Under non-normal residuals, empirical power increased with sample size but differed according to the form of departure from normality. The Shapiro–Francia and Energy tests showed the strongest performance under exponential residuals, while the Anderson–Darling test performed best under heavy-tailed and contaminated normal residuals. The Random Projection test achieved the highest empirical power under bimodal residuals. The Lilliefors test generally showed the lowest empirical power. These findings indicate that the relative performance of normality tests depends on residual distributional characteristics and sample size, supporting context-specific test selection in regression diagnostics.