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August 2007 Aggregation for Gaussian regression
Florentina Bunea, Alexandre B. Tsybakov, Marten H. Wegkamp
Ann. Statist. 35(4): 1674-1697 (August 2007). DOI: 10.1214/009053606000001587

Abstract

This paper studies statistical aggregation procedures in the regression setting. A motivating factor is the existence of many different methods of estimation, leading to possibly competing estimators. We consider here three different types of aggregation: model selection (MS) aggregation, convex (C) aggregation and linear (L) aggregation. The objective of (MS) is to select the optimal single estimator from the list; that of (C) is to select the optimal convex combination of the given estimators; and that of (L) is to select the optimal linear combination of the given estimators. We are interested in evaluating the rates of convergence of the excess risks of the estimators obtained by these procedures. Our approach is motivated by recently published minimax results [Nemirovski, A. (2000). Topics in non-parametric statistics. Lectures on Probability Theory and Statistics (Saint-Flour, 1998). Lecture Notes in Math. 1738 85–277. Springer, Berlin; Tsybakov, A. B. (2003). Optimal rates of aggregation. Learning Theory and Kernel Machines. Lecture Notes in Artificial Intelligence 2777 303–313. Springer, Heidelberg]. There exist competing aggregation procedures achieving optimal convergence rates for each of the (MS), (C) and (L) cases separately. Since these procedures are not directly comparable with each other, we suggest an alternative solution. We prove that all three optimal rates, as well as those for the newly introduced (S) aggregation (subset selection), are nearly achieved via a single “universal” aggregation procedure. The procedure consists of mixing the initial estimators with weights obtained by penalized least squares. Two different penalties are considered: one of them is of the BIC type, the second one is a data-dependent $\ell_1$-type penalty.

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Florentina Bunea. Alexandre B. Tsybakov. Marten H. Wegkamp. "Aggregation for Gaussian regression." Ann. Statist. 35 (4) 1674 - 1697, August 2007. https://doi.org/10.1214/009053606000001587

Information

Published: August 2007
First available in Project Euclid: 29 August 2007

zbMATH: 1209.62065
MathSciNet: MR2351101
Digital Object Identifier: 10.1214/009053606000001587

Subjects:
Primary: 62G08
Secondary: 62C20 , 62G05 , 62G20

Keywords: Aggregation , Lasso estimator , minimax risk , model averaging , Model selection , Nonparametric regression , Oracle inequalities , penalized least squares

Rights: Copyright © 2007 Institute of Mathematical Statistics

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Vol.35 • No. 4 • August 2007
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