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Statistica Sinica 19 (2009), 1281-1296



Lan Xue

Oregon State University

Abstract: We propose a penalized polynomial spline method for simultaneous model estimation and variable selection in additive models. It approximates nonparametric functions by polynomial splines, and minimizes the sum of squared errors subject to an additive penalty on norms of spline functions. This approach sets estimators of certain function components to zero, thus performing variable selection. Under mild conditions, we show that the newly proposed method estimates the non-zero function components in the model with the same optimal mean square convergence rate as the standard polynomial spline estimators, and correctly sets the zero function components to zero with probability approaching one, as $n$ goes to infinity. Besides being theoretically justified, the proposed method is easy to understand and straightforward to implement. Extensive Monte Carlo simulation studies show the newly proposed method compares favorably with the existing ones in finite sample performance. We also illustrate the use of the proposed method by analyzing two data sets.

Key words and phrases: Boston housing price, knot, mean square consistency, ozone data, penalized least squares, SCAD.

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