A robust and efficient estimation and variable selection method for partially linear single-index models

• We propose a robust and efficient estimation method based on local modal regression.
• The asymptotic normality of the proposed estimators are established.
• The estimator is shown to be superior compared with OLS.
• A variable selection procedure for covariates is shown to possess oracle property.
• A modified EM algorithm is proposed.

In this paper, a new robust and efficient estimation approach based on local modal regression is proposed for partially linear single-index models, of which the univariate nonparametric link function is approximated by local polynomial regression. The asymptotic normality of proposed estimators for both the parametric and nonparametric parts are established. We show that the resulting estimator is more efficient than the ordinary least-square-based estimation in the case of outliers or heavy-tail error distribution, and as asymptotically efficient as the least square estimator when there are no outliers and the error is normal distribution. To achieve sparsity when there exist irrelevant variables in the model, a variable selection procedure based on SCAD penalty is developed to select significant parametric covariates and is shown to possess oracle property under some regularity conditions. We also propose a practical modified EM algorithm for the new method. Some Monte Carlo simulations and a real data set are conducted to illustrate the finite sample performance of the proposed estimators.