Outlier-resistant high-dimensional regression modelling based on distribution-free outlier detection and tuning parameter selection

The L-1-type regularization is a useful tool for high-dimensional regression modelling. Although the L-1-type approaches perform well regression modelling, the methods suffer from outliers, since the L-1-type approaches are based on non-robust methods (e. g. least squares loss function). In order to resolve the drawback, we propose a robust L-1-type regularization method based on distribution-free outlier detection measure. We consider outlier detection in principal component spaces (PCSs) to overcome dimensionality problem of high-dimensional data, and propose a novel cut-off value based on a non-parametric test. By using the distribution-free outlier detection measure, we can effectively detect outliers in PCS without distribution assumption of the Mahalanobis distance. We then propose a robust L-1-type regularization method via a weighted elastic net. The tuning parameter selection is a vital matter in L-1-type regularized regression modelling, since choosing the tuning parameters can be seen as variable selection and model estimation. We derive an information criterion to select the tuning parameters of the proposed robust L-1-type regularization method. Monte Carlo simulations and NCI60 data analysis show that the proposed robust regression modelling strategies effectively perform for high-dimensional regression modelling, even in the presence of outliers.