When we consider regression problems, there exist two approaches: the parametric approach with a suitably chosen model including a finite dimensional parameter vector, and the nonparametric approach without any assumption of structure. The nonparametric approach has been known to be useful for regression problem, which generally has consistency as an asymptotic property. However the estimated structure with nonparametric approach is not easy to understand. The parametric approach, on the other hand, does provide a nice interpretation of the estimated structure,
though it includes structural bias in general. We propose a semiparametric penalized spline estimator which has a parametric structure as an initial guess. The residual of the parametric model is estimated nonparametrically by penalized spline method. The interpretation of estimated structure by the proposed semiparametric estimator becomes clearer than that by fully nonparametric estimator. Asymptotic theory for the proposed semiparametric estimator is developed, which shows that its behavior is depending on the asymptotic property of fully nonparametric penalized spline estimator as well as the discrepancy between the true regression function and the parametric part.
As a naturally associated application of asymptotics, some criteria for the selection of parametric models are addressed. Numerical experiments show that the proposed estimator performs better than kernel-based existing
semiparametric estimator and fully nonparametric estimator, and the proposed criteria work well for choosing a reasonable parametric model.