Robust estimation for partially linear single-index model

In this article, we investigate a new estimation approach for the partially linear single-index model based on modal regression method, where the non parametric function is estimated by penalized spline method. Moreover, we develop an expection maximum (EM)-type algorithm and establish the large sample properties of the proposed estimation method. A distinguishing characteristic of the newly proposed estimation is robust against outliers through introducing an additional tuning parameter which can be automatically selected using the observed data. Simulation studies and real data example are used to evaluate the finite-sample performance, and the results show that the newly proposed method works very well.