Open Access
December 2009 Wavelet regression in random design with heteroscedastic dependent errors
Rafał Kulik, Marc Raimondo
Ann. Statist. 37(6A): 3396-3430 (December 2009). DOI: 10.1214/09-AOS684


We investigate function estimation in nonparametric regression models with random design and heteroscedastic correlated noise. Adaptive properties of warped wavelet nonlinear approximations are studied over a wide range of Besov scales, $f\in\mathcal{B}_{\pi,r}^{s}$, and for a variety of Lp error measures. We consider error distributions with Long-Range-Dependence parameter α, 0<α≤1; heteroscedasticity is modeled with a design dependent function σ. We prescribe a tuning paradigm, under which warped wavelet estimation achieves partial or full adaptivity results with the rates that are shown to be the minimax rates of convergence. For p>2, it is seen that there are three rate phases, namely the dense, sparse and long range dependence phase, depending on the relative values of s, p, π and α. Furthermore, we show that long range dependence does not come into play for shape estimation f∫f. The theory is illustrated with some numerical examples.


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Rafał Kulik. Marc Raimondo. "Wavelet regression in random design with heteroscedastic dependent errors." Ann. Statist. 37 (6A) 3396 - 3430, December 2009.


Published: December 2009
First available in Project Euclid: 17 August 2009

zbMATH: 1369.62074
MathSciNet: MR2549564
Digital Object Identifier: 10.1214/09-AOS684

Primary: 62G05
Secondary: 62G08 , 62G20

Keywords: adaptive estimation , Long range dependence , maxiset , Nonparametric regression , random design , shape estimation , thresholding , warped wavelets , Wavelets

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 6A • December 2009
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