Robust regression shrinkage and consistent variable selection through the LAD-lasso

The least absolute deviation (LAD) regression is a useful method for robust regression, and the least absolute shrinkage and selection operator (lasso) is a popular choice for shrinkage estimation and variable selection. In this article we combine these two classical ideas together to produce LAD-lasso. Compared with the LAD regression, LAD-lasso can do parameter estimation and variable selection simultaneously. Compared with the traditional lasso, LAD-lasso is resistant to heavy-tailed errors or outliers in the response. Furthermore, with easily estimated tuning parameters, the LAD-lasso estimator enjoys the same asymptotic efficiency as the unpenalized LAD estimator obtained under the true model (i.e., the oracle property). Extensive simulation studies demonstrate satisfactory finite-sample performance of LAD-lasso, and a real example is analyzed for illustration purposes.