Surface estimation, variable selection, and the nonparametric oracle property

Variable selection for multivariate nonparametric regression is an impor-tant, yet challenging, problem due, in part, to the infinite dimensionality of the function space. An ideal selection procedure should be automatic, stable, easy to use, and have desirable asymptotic properties. In particular, we define a selection procedure to be nonparametric oracle (np-oracle) if it consistently se-lects the correct subset of predictors and at the same time estimates the smooth surface at the optimal nonparametric rate, as the sample size goes to infinity. In this paper, we propose a model selection procedure for nonparametric mod-els, and explore the conditions under which that the new method enjoys the aforementioned properties. Developed in the framework of smoothing spline ANOVA, our estimator is obtained via solving a regularization problem with a novel adaptive penalty on the sum of functional component norms. Theo-retical properties of the new estimator are established. Additionally, numerous simulated and real examples further demonstrate that the new approach sub-stantially outperforms other existing methods in the finite sample setting