  We consider model selection and estimation for partial spline models and
propose a new regularization method in the context of smoothing splines. The
regularization method has a simple yet elegant form, consisting of roughness
penalty on the nonparametric component and shrinkage penalty on the parametric
components, which can achieve function smoothing and sparse estimation
simultaneously. We establish the convergence rate and oracle properties of the
estimator under weak regularity conditions. Remarkably, the estimated
parametric components are sparse and efficient, and the nonparametric component
can be estimated with the optimal rate. The procedure also has attractive
computational properties. Using the representer theory of smoothing splines, we
reformulate the objective function as a LASSO-type problem, enabling us to use
the LARS algorithm to compute the solution path. We then extend the procedure
to situations when the number of predictors increases with the sample size and
investigate its asymptotic properties in that context. Finite-sample
performance is illustrated by simulations.
