Sieve M-estimation for semiparametric varying-coefficient partially linear regression model

This article considers a semiparametric varying-coefficient partially linear regression model. The semiparametric varying-coefficient partially linear regression model which is a generalization of the partially linear regression model and varying-coefficient regression model that allows one to explore the possibly nonlinear effect of a certain covariate on the response variable. A sieve M-estimation method is proposed and the asymptotic properties of the proposed estimators are discussed. Our main object is to estimate the nonparametric component and the unknown parameters simultaneously. It is easier to compute and the required computation burden is much less than the existing two-stage estimation method. Furthermore, the sieve M-estimation is robust in the presence of outliers if we choose appropriate ρ(·). Under some mild conditions, the estimators are shown to be strongly consistent; the convergence rate of the estimator for the unknown nonparametric component is obtained and the estimator for the unknown parameter is shown to be asymptotically normally distributed. Numerical experiments are carried out to investigate the performance of the proposed method.