Estimation and inference for additive partially nonlinear models

In this paper, we extend the additive partially linear model to the additive partially nonlinear model in which the linear part of the additive partially linear model is replaced by a nonlinear function of the covariates. A profile nonlinear least squares estimation procedure for the parameter vector in nonlinear function and the nonparametric functions of the additive partially nonlinear model is proposed and the asymptotic properties of the resulting estimators are established. Furthermore, we apply the empirical likelihood method to the additive partially nonlinear model. An empirical likelihood ratio for the parameter vector and a residual adjusted empirical likelihood ratio for the nonparametric functions have been proposed. Wilks phenomenon is proved and the confidence regions for the parametric vector and the nonparametric functions are constructed. Some simulations have been conducted to assess the performance of the proposed estimating procedures. The results have demonstrated that both the procedures perform well in finite samples. Compared with the results from the empirical likelihood method with those from the profile nonlinear least squares method, the empirical likelihood method performs better in terms of coverage probabilities and average widths of confidence bands.