Testing normality in the presence of outliers

Statistical models are often based on normal distributions and procedures for testing this distributional assumption are needed. Many goodness-of-fit tests suffer from the presence of outliers, in the sense that they may reject the null hypothesis even in the case of a single extreme observation. We show a possible extension of the Shapiro-Wilk test that is not affected by such a problem. The presented method is inspired by the forward search (FS), a new, recently proposed, diagnostic tool. An application to univariate observations shows how the procedure is able to capture the structure of the data, even in the presence of outliers. Other properties are also investigated.