Abstract

Most existing methods for single-index models (SIM) were focused on either mean regression or quantile regression, while the former is sensitive to outliers or heavy tailed distributions and the latter suffers from efficiency and uniqueness of estimation. In this paper, we develop a robust, efficient and easily implemented estimation procedure for index coefficient by integrating the ideas of local modal regression and outer product gradients. Under some mild condition, we establish the asymptotic normality of proposed estimators, and further discuss the optimal choices of tuning parameters, including one common bandwidth for nonparametric polynomial smoothing and another key bandwidth that controls the robustness and efficiency of the estimator, based on the derived theories. A practical modified EM algorithm is also presented for implementation. Finally, some simulation studies and a Boston housing price data are conducted to confirm the merits and theoretical findings of the novel method.