The adaptive L1-penalized LAD regression for partially linear single-index models

The penalized least square (LS) method has been recently studied as a popular technique for simultaneous estimation and variable selection in partially linear single-index models (PLSIMs). However, an LS estimator may lose its superiorities if there exist outliers in the response variables or the error is heavy-tailed distributed, and the least absolute deviation (LAD) regression is a useful method in this case. In this paper, we propose a stepwise penalized LAD regression to generate robust estimators based on PLSIM. An iterative procedure is firstly presented to estimate the index parameters with the univariate link function approximated by local linear LAD regression, then an adaptive L1-penalized LAD procedure is introduced to do estimation and variable selection for the linear part parameters based on the index estimator. Compared with the penalized LS estimator, our proposed estimator is resistant to heavy-tailed errors or outliers in the response. Furthermore, under some suitable conditions, the theoretical properties including asymptotic normality of the index parametric estimator and oracle property of the linear parametric estimator are established. Some Monte Carlo simulations and a real data set are conducted to illustrate the finite sample performance of the estimators.