  In this paper, we propose a new semiparametric regression estimator by using
a hybrid technique of a parametric approach and a nonparametric penalized
spline method. The overall shape of the true regression function is captured by
the parametric part, while its residual is consistently estimated by the
nonparametric part. Asymptotic theory for the proposed semiparametric estimator
is developed, showing that its behavior is dependent on the asymptotics for the
nonparametric penalized spline estimator as well as on 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 the existing kernel-based semiparametric estimator and the
fully nonparametric estimator, and that the proposed criteria work well for
choosing a reasonable parametric model.
