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Surface estimation, variable selection, and the nonparametric oracle property. (English) Zbl 1214.62044
Summary: Variable selection for multivariate nonparametric regression is an important, yet challenging, problem due, in part, to the infinite dimensionality of the function space. An ideal selection procedure would 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 selects 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. We propose a model selection procedure for nonparametric models, and explore the conditions under which 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. Theoretical properties of the new estimator are established. Additionally, numerous simulations and examples suggest that the new approach substantially outperforms other existing methods in the finite sample setting.

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62J10 Analysis of variance and covariance (ANOVA)
65C60 Computational problems in statistics (MSC2010)
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