On the adaptive wavelet estimation of a multidimensional regression function under α -mixing dependence: Beyond the standard assumptions on the noise

Christophe Chesneau

Commentationes Mathematicae Universitatis Carolinae (2013)

  • Volume: 54, Issue: 4, page 527-556
  • ISSN: 0010-2628

Abstract

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We investigate the estimation of a multidimensional regression function f from n observations of an α -mixing process ( Y , X ) , where Y = f ( X ) + ξ , X represents the design and ξ the noise. We concentrate on wavelet methods. In most papers considering this problem, either the proposed wavelet estimator is not adaptive (i.e., it depends on the knowledge of the smoothness of f in its construction) or it is supposed that ξ is bounded or/and has a known distribution. In this paper, we go far beyond this classical framework. Under no boundedness assumption on ξ and no a priori knowledge on its distribution, we construct adaptive term-by-term thresholding wavelet estimators attaining “sharp” rates of convergence under the mean integrated squared error over a wide class of functions f .

How to cite

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Chesneau, Christophe. "On the adaptive wavelet estimation of a multidimensional regression function under $\alpha $-mixing dependence: Beyond the standard assumptions on the noise." Commentationes Mathematicae Universitatis Carolinae 54.4 (2013): 527-556. <http://eudml.org/doc/260751>.

@article{Chesneau2013,
abstract = {We investigate the estimation of a multidimensional regression function $f$ from $n$ observations of an $\alpha $-mixing process $(Y,X)$, where $Y=f(X)+\xi $, $X$ represents the design and $\xi $ the noise. We concentrate on wavelet methods. In most papers considering this problem, either the proposed wavelet estimator is not adaptive (i.e., it depends on the knowledge of the smoothness of $f$ in its construction) or it is supposed that $\xi $ is bounded or/and has a known distribution. In this paper, we go far beyond this classical framework. Under no boundedness assumption on $\xi $ and no a priori knowledge on its distribution, we construct adaptive term-by-term thresholding wavelet estimators attaining “sharp” rates of convergence under the mean integrated squared error over a wide class of functions $f$.},
author = {Chesneau, Christophe},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {nonparametric regression; $\alpha $-mixing dependence; adaptive estimation; wavelet methods; rates of convergence; nonparametric regression; -mixing dependence; wavelet methods; rate of convergence; adaptive estimation},
language = {eng},
number = {4},
pages = {527-556},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the adaptive wavelet estimation of a multidimensional regression function under $\alpha $-mixing dependence: Beyond the standard assumptions on the noise},
url = {http://eudml.org/doc/260751},
volume = {54},
year = {2013},
}

TY - JOUR
AU - Chesneau, Christophe
TI - On the adaptive wavelet estimation of a multidimensional regression function under $\alpha $-mixing dependence: Beyond the standard assumptions on the noise
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2013
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 54
IS - 4
SP - 527
EP - 556
AB - We investigate the estimation of a multidimensional regression function $f$ from $n$ observations of an $\alpha $-mixing process $(Y,X)$, where $Y=f(X)+\xi $, $X$ represents the design and $\xi $ the noise. We concentrate on wavelet methods. In most papers considering this problem, either the proposed wavelet estimator is not adaptive (i.e., it depends on the knowledge of the smoothness of $f$ in its construction) or it is supposed that $\xi $ is bounded or/and has a known distribution. In this paper, we go far beyond this classical framework. Under no boundedness assumption on $\xi $ and no a priori knowledge on its distribution, we construct adaptive term-by-term thresholding wavelet estimators attaining “sharp” rates of convergence under the mean integrated squared error over a wide class of functions $f$.
LA - eng
KW - nonparametric regression; $\alpha $-mixing dependence; adaptive estimation; wavelet methods; rates of convergence; nonparametric regression; -mixing dependence; wavelet methods; rate of convergence; adaptive estimation
UR - http://eudml.org/doc/260751
ER -

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