Building a robust linear model with forward selection and stepwise procedures

Classical step-by-step algorithms, such as forward selection (FS) and stepwise (SW) methods, are computationally suitable, but yield poor results when the data contain outliers and other contaminations. Robust model selection procedures, on the other hand, are not computationally efficient or scalable to large dimensions, because they require the fitting of a large number of submodels. Robust and computationally efficient versions of FS and SW are proposed. Since FS and SW can be expressed in terms of sample correlations, simple robustifications are obtained by replacing these correlations by their robust counterparts. A pairwise approach is used to construct the robust correlation matrix—not only because of its computational advantages over the d-dimensional approach, but also because the pairwise approach is more consistent with the idea of step-by-step algorithms. The proposed robust methods have much better performance compared to standard FS and SW. Also, they are computationally very suitable and scalable to large high-dimensional data sets.