Statistical inference on semi-parametric partial linear additive models

In the framework of partial linear additive models, we first develop a profile least-squares estimation of the parametric component based on Liang et al.'s [(2008), 'Additive Partial Linear Models with Measurement Errors', Biometrika, 95(3), 667-678] work. This estimator is shown to be asymptotically normal and root-n consistent without requirement of undersmoothing of the nonparametric component. Next, when some additional linear restrictions on the parametric component are available, we postulate a restricted profile least-squares estimator for the parametric component and prove the asymptotic normality of the resulting estimator. To check the validity of the linear constraints on the parametric component, we explore a generalised likelihood ratio test statistic and demonstrate that it follows asymptotically chi-squared distribution under the null hypothesis. Thus, the result unveils a new Wilks type of phenomenon. Simulation studies are conducted to illustrate the proposed methods. An application to the crime rate data in Columbus (Ohio) has been carried out.