Smoothing spline in semiparametric additive regression model with Bayesian approach

Semiparametric additive regression model is a combination of parametric and nonparametric regression models. The parametric components are not linear but following a polynomial pattern, while the nonparametric components are unknown pattern and assumed to be contained in the Sobolev space. The nonparametric components can be approximated by smoothing spline functions. In the development of smoothing spline, the classical statistical approach cannot be applied for solving the inference problem such as constructing confidence intervals for the regression curve. To construct confidence interval of smoothing spline curve in the semiparametric additive regression model, we propose to use Bayesian approach, by assuming improper Gaussian distribution for prior distribution in nonparametric components and multivariate normal distribution for parametric components. In this study, we obtain parameter estimators for parametric component and smoothing spline estimators for the nonparametric component in semiparametric additive regression model. Moreover, we also develop a smoothing parameters selection method simultaneously using Generalized Maximum Likelihood (GML) and confidence intervals for the parameters of the parametric component and the smoothing spline functions of the nonparametric component using Bayesian approach. By computing each posterior mean and posterior variance of parametric component parameters and smoothing spline functions, confidence intervals can be constructed for the parametric component parameters and confidence interval smoothing spline functions for nonparametric components in semiparametric additive regression models. We create R-code to implement estimation model and inference procedure. Our simulation studies reveal estimation and inference method perform reasonably well.