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August 2015 Bandwidth selection in kernel empirical risk minimization via the gradient
Michaël Chichignoud, Sébastien Loustau
Ann. Statist. 43(4): 1617-1646 (August 2015). DOI: 10.1214/15-AOS1318

Abstract

In this paper, we deal with the data-driven selection of multidimensional and possibly anisotropic bandwidths in the general framework of kernel empirical risk minimization. We propose a universal selection rule, which leads to optimal adaptive results in a large variety of statistical models such as nonparametric robust regression and statistical learning with errors in variables. These results are stated in the context of smooth loss functions, where the gradient of the risk appears as a good criterion to measure the performance of our estimators. The selection rule consists of a comparison of gradient empirical risks. It can be viewed as a nontrivial improvement of the so-called Goldenshluger–Lepski method to nonlinear estimators. Furthermore, one main advantage of our selection rule is the nondependency on the Hessian matrix of the risk, usually involved in standard adaptive procedures.

Citation

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Michaël Chichignoud. Sébastien Loustau. "Bandwidth selection in kernel empirical risk minimization via the gradient." Ann. Statist. 43 (4) 1617 - 1646, August 2015. https://doi.org/10.1214/15-AOS1318

Information

Received: 1 January 2014; Revised: 1 January 2015; Published: August 2015
First available in Project Euclid: 17 June 2015

zbMATH: 1317.62026
MathSciNet: MR3357873
Digital Object Identifier: 10.1214/15-AOS1318

Subjects:
Primary: 62G05, 62G20
Secondary: 62G08, 62H30

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.43 • No. 4 • August 2015
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