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October 2014 Adaptive function estimation in nonparametric regression with one-sided errors
Moritz Jirak, Alexander Meister, Markus Reiß
Ann. Statist. 42(5): 1970-2002 (October 2014). DOI: 10.1214/14-AOS1248

Abstract

We consider the model of nonregular nonparametric regression where smoothness constraints are imposed on the regression function $f$ and the regression errors are assumed to decay with some sharpness level at their endpoints. The aim of this paper is to construct an adaptive estimator for the regression function $f$. In contrast to the standard model where local averaging is fruitful, the nonregular conditions require a substantial different treatment based on local extreme values. We study this model under the realistic setting in which both the smoothness degree $\beta>0$ and the sharpness degree $\mathfrak{a}\in(0,\infty)$ are unknown in advance. We construct adaptation procedures applying a nested version of Lepski’s method and the negative Hill estimator which show no loss in the convergence rates with respect to the general $L_{q}$-risk and a logarithmic loss with respect to the pointwise risk. Optimality of these rates is proved for $\mathfrak{a}\in(0,\infty)$. Some numerical simulations and an application to real data are provided.

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Moritz Jirak. Alexander Meister. Markus Reiß. "Adaptive function estimation in nonparametric regression with one-sided errors." Ann. Statist. 42 (5) 1970 - 2002, October 2014. https://doi.org/10.1214/14-AOS1248

Information

Published: October 2014
First available in Project Euclid: 11 September 2014

zbMATH: 1305.62172
MathSciNet: MR3262474
Digital Object Identifier: 10.1214/14-AOS1248

Subjects:
Primary: 62G08 , 62G32

Keywords: Adaptive convergence rates , Bandwidth selection , frontier estimation , Lepski’s method , Minimax optimality , negative Hill estimator , nonregular regression

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.42 • No. 5 • October 2014
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