Electronic Journal of Statistics

Optimal upper and lower bounds for the true and empirical excess risks in heteroscedastic least-squares regression

Adrien Saumard

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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.

Article information

Source
Electron. J. Statist., Volume 6 (2012), 579-655.

Dates
First available in Project Euclid: 18 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1334754008

Digital Object Identifier
doi:10.1214/12-EJS679

Mathematical Reviews number (MathSciNet)
MR2988421

Zentralblatt MATH identifier
1334.62068

Keywords
Least-squares regression heteroscedasticity excess risk lower bounds sup-norm localized basis empirical process

Citation

Saumard, Adrien. Optimal upper and lower bounds for the true and empirical excess risks in heteroscedastic least-squares regression. Electron. J. Statist. 6 (2012), 579--655. doi:10.1214/12-EJS679. https://projecteuclid.org/euclid.ejs/1334754008


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