Robust Signed-Rank Variable Selection in Linear Regression

The growing need for dealing with big data has made it necessary to find computationally efficient methods for identifying important factors to be considered in statistical modeling. In the linear model, the Lasso is an effective way of selecting variables using penalized regression. It has spawned substantial research in the area of variable selection for models that depend on a linear combination of predictors. However, work addressing the lack of optimality of variable selection when the model errors are not Gaussian and/or when the data contain gross outliers is scarce. We propose the weighted signed-rank Lasso as a robust and efficient alternative to least absolute deviations and least squares Lasso. The approach is appealing for use with big data since one can use data augmentation to perform the estimation as a single weighted L1 optimization problem. Selection and estimation consistency are theoretically established and evaluated via simulation studies. The results confirm the optimality of the rank-based approach for data with heavy-tailed and contaminated errors or data containing high-leverage points.