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We consider model selection and estimation for partial spline models andpropose a new regularization method in the context of smoothing splines. Theregularization method has a simple yet elegant form, consisting of roughnesspenalty on the nonparametric component and shrinkage penalty on the parametriccomponents, which can achieve function smoothing and sparse estimationsimultaneously. We establish the convergence rate and oracle properties of theestimator under weak regularity conditions. Remarkably, the estimatedparametric components are sparse and efficient, and the nonparametric componentcan be estimated with the optimal rate. The procedure also has attractivecomputational properties. Using the representer theory of smoothing splines, wereformulate the objective function as a LASSO-type problem, enabling us to usethe LARS algorithm to compute the solution path. We then extend the procedureto situations when the number of predictors increases with the sample size andinvestigate its asymptotic properties in that context. Finite-sampleperformance is illustrated by simulations.

Sparse and Efficient Estimation for Partial Spline Models with Increasing Dimension