New robust LASSO method based on ranks

The LASSO (Least Absolute Shrinkage and Selection Operator) has been a popular technique for simultaneous linear regression estimation and variable selection. Robust approaches for LASSO are needed in the case of heavy-tailed errors or severe outliers. We propose a novel robust LASSO method that has a non-parametric flavor: it solves a criterion function based on ranks of the residuals with LASSO penalty. The criterion is based on pairwise differences of residuals in the least absolute deviation (LAD) loss leading to a bounded influence function. With the i\-criterion we can easily incorporate other penalties such as fused LASSO for group sparsity and smoothness. For both methods, we propose efficient algorithms for computing the solutions. Our simulation study and application examples (image denoising, prostate cancer data analysis) show that our method outperform the usual LS/LASSO methods for either heavy-tailed errors or outliers, offering better variable selection than another robust competitor, LAD-LASSO method.