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2012 Optimal upper and lower bounds for the true and empirical excess risks in heteroscedastic least-squares regression
Adrien Saumard
Electron. J. Statist. 6(none): 579-655 (2012). DOI: 10.1214/12-EJS679

Abstract

We consider the estimation of a bounded regression function with nonparametric heteroscedastic noise and random design. We study the true and empirical excess risks of the least-squares estimator on finite-dimensional vector spaces. We give upper and lower bounds on these quantities that are nonasymptotic and optimal to first order, allowing the dimension to depend on sample size. These bounds show the equivalence between the true and empirical excess risks when, among other things, the least-squares estimator is consistent in sup-norm with the projection of the regression function onto the considered model. Consistency in the sup-norm is then proved for suitable histogram models and more general models of piecewise polynomials that are endowed with a localized basis structure.

Citation

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Adrien Saumard. "Optimal upper and lower bounds for the true and empirical excess risks in heteroscedastic least-squares regression." Electron. J. Statist. 6 579 - 655, 2012. https://doi.org/10.1214/12-EJS679

Information

Published: 2012
First available in Project Euclid: 18 April 2012

zbMATH: 1334.62068
MathSciNet: MR2988421
Digital Object Identifier: 10.1214/12-EJS679

Rights: Copyright © 2012 The Institute of Mathematical Statistics and the Bernoulli Society

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