Wavelet regression in random design with heteroscedastic dependent errors

TL;DR

This paper develops adaptive wavelet-based methods for nonparametric function estimation in heteroscedastic, correlated noise settings, achieving minimax rates across various Besov spaces and error measures, with special insights into long-range dependence effects.

Contribution

It introduces a tuning paradigm for warped wavelet estimators that attain minimax rates in heteroscedastic, dependent noise models across diverse Besov spaces, including phase distinctions based on dependence and sparsity.

Findings

Wavelet estimators achieve minimax rates under heteroscedastic, dependent noise.

Three rate phases identified: dense, sparse, and long-range dependence.

Long-range dependence does not affect shape estimation accuracy.

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 $\alpha,0<\alpha\leq1$; 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.