Wavelet regression in random design with heteroscedastic dependent errors

Abstract

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 $L^p$ error measures. We consider error distributions with Long-Range-Dependence parameter $\alpha ,0<\alpha \le 1$; heteroscedasticity is modeled with a design dependent function $\sigma$. 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 $s,p,\pi$ and $\alpha$. Furthermore, we show that long range dependence does not come into play for shape estimation $f-\int f$. The theory is illustrated with some numerical examples.

Published in at http://dx.doi.org/10.1214/09-AOS684 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Functional deconvolution in a periodic setting: Uniform case
• Wavelet regression in random design with heteroscedastic dependent errors
• Asymptotic inference in some heteroscedastic regression models with long memory design and errors

Cited by in corpus (1)

• Wavelet regression in random design with heteroscedastic dependent errors