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Goodness-of-Fit Test of Shapiro-Wilk Type with Nuisance Regression and Scale

Shapiro and Wilk (1965) proposed a highly intuitive goodness-of-fit test of normality with nuisance location and scale parameters. The test has received a considerable attention in the literature; its asymptotic null distribution is covered by the results of de Wet and Wenter (1973), and was recently studied by Sen (2002). We extend the Shapiro-Wilk test to the situation with nuisance regression and scale, and construct a test based on the pair of the maximum likelihood estimator and of a pseudo-L-estimator of the standard deviation in the linear regression model. The asymptotic equivalence of these estimators is a characteristic property of the normal distribution of the errors. We shall show that the asymptotic null distribution of the test criterion under the hypothesis of normality is similar to that of the Shapiro-Wilk test in the location-scale model. The main tool of the proof is the second-order asymptotics of L-estimators (see Jureˇckov´a and Sen, 1996) extended to pseudo-L-estimators in regression model. The proposed test is numerically compared with a test earlier proposed by the authors (see Jureˇckov´a et al., 2003), based on robust estimators of scale.