Shapiro–Wilk-type test of normality under nuisance regression and scale

An extension of the Shapiro–Wilk test to verify the hypothesis of normality in the presence of nuisance regression and scale has been previously considered. Such a test is typically based on the pair of the maximum likelihood and BLUE estimators of the standard deviation in the linear regression model. It has been shown that the asymptotic null distribution of the test criterion, extended to the regression model, is equivalent to that of the original Shapiro–Wilk test for the location-scale model. A simulation study is performed in order to show that both criteria are close under the normality hypothesis for moderate as well for large data sets. The power of the test against various alternative distributions of the model errors is illustrated. Furthermore, it is shown that the probabilities of errors of both the first and second kinds do not depend on the design matrix or on the parameters of the linear model.