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, $f\in\mathcal{B}^s_{\pi,r}$, and for a variety of $L^p$ error measures. We consider error distributions with Long-Range-Dependence parameter $\alpha,0 2$, 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.