Two-stage estimation and simultaneous confidence band in partially nonlinear additive model

In this paper, we focus on the estimation and inference in partially nonlinear additive model on which few research was conducted to our best knowledge. By integrating spline approximation and local smoothing, we propose a two-stage estimating approach in which the profile nonlinear least square method was used to estimate parameters and additive functions. Under some regular conditions, we establish the asymptotic normality of parametric estimators and achieve an optimal nonparametric convergence rate of the fitted functions. Furthermore, the spline-backfitted local linear estimator is proposed for the additive functions and the corresponding asymptotic distribution is also established. To make inference on the nonparametric functions from the whole, we construct the theoretical simultaneous confidence bands, and further propose an empirical bootstrap-based confidence band for the heavy computing burden in implement. Finally, both Monte Carlo simulation and real data analysis show the good performance of our proposed methods.