Asymptotic analysis of the Huberized LASSO estimator

The Huberized LASSO model, a robust version of the popular LASSO, yields robust model selection in sparse linear regression. Though its superior performance was empirically demonstrated for large variance noise, currently no theoretical asymptotic analysis has been derived for the Huberized LASSO estimator. Here we prove that the Huberized LASSO estimator is consistent and asymptotically normal distributed under a proper shrinkage rate. Our derivation shows that, unlike the LASSO estimator, its asymptotic variance is stabilized in the presence of noise with large variance. We also propose the adaptive Huberized LASSO estimator by allowing unequal penalty weights for the regression coefficients, and prove its model selection consistency. Simulations confirm our theoretical results.