Inferences for extended partially linear single-index models

In partially linear single-index models, there are two different covariate matrices in the model for the linear part and nonlinear part. All covariate information needs to be divided into two parts before the model can be fitted. In contrast, in the extended partially linear single-index models, all the covariate variables are included in one matrix, which is contained in both the linear part and nonlinear part of the model. We propose local smoothing estimators for the model parameters and unknown function with computationally efficient and accurate computation methodologies and obtain the asymptotic properties of the model parameter estimators. We also employ the LASSO penalty to obtain penalized estimators with consistency and oracle property in order to carry out estimation and variable selection simultaneously. Then we develop a linear hypothesis test for the model parameters. Furthermore, we extend the proposed methodology to the increasing dimensional settings under certain assumptions. Simulation studies are presented that support our analytic results. In addition, a real data analysis is provided for illustration.