Single-index modal regression via outer product gradients

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 may lose efficiency for normally distributed data. Then a robust, efficient and easily implemented estimation procedure for index coefficient in SIM is developed by integrating the ideas of local modal regression and outer product gradients. Under some mild conditions, we establish the asymptotic normality of the proposed estimators. We 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 two real data analysis are conducted to confirm the merits and theoretical findings of the novel method.