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.