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Statistica Sinica 21 (2011), 679-705



Curtis B. Storlie$^1$, Howard D. Bondell$^2$, Brian J. Reich$^2$
and Hao Helen Zhang$^2$

$^1$University of New Mexico and $^2$North Carolina State University

Abstract: 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. In this paper, 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.

Key words and phrases: Adaptive LASSO, nonparametric regression, regularization method, smoothing spline ANOVA, variable selection.

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