Statistical inference for partially linear regression models with measurement errors

In this paper, the authors investigate three aspects of statistical inference for the partially linear regression models where some covariates are measured with errors. Firstly, a bandwidth selection procedure is proposed, which is a combination of the difference-based technique and GCV method. Secondly, a goodness-of-fit test procedure is proposed, which is an extension of the generalized likelihood technique. Thirdly, a variable selection procedure for the parametric part is provided based on the nonconcave penalization and corrected profile least squares. Same as “Variable selection via nonconcave penalized likelihood and its oracle properties” (J. Amer. Statist. Assoc., 96, 2001, 1348–1360), it is shown that the resulting estimator has an oracle property with a proper choice of regularization parameters and penalty function. Simulation studies are conducted to illustrate the finite sample performances of the proposed procedures.