L^p-wavelet regression with correlated errors and inverse problems

Rafal Kułik and Marc Raimondo

Abstract

We investigate global performances of non-linear wavelet estimation in regression models with correlated errors. Convergence properties are studied over a wide range of Besov classes \(\mathcal{B}^s_{\pi,r}\) and for a variety of \(L^p\) error measures. We consider error distributions with Long-Range-Dependence parameter \(\alpha\), \(0 < \alpha \le 1\). In this setting we present a single adaptive wavelet thresholding estimator which achieves near-optimal properties simultaneously over a class of spaces and error measures. Our method reveals an elbow feature in the rate of convergence at \(s= \frac{\alpha}{2}(\frac{p}{\pi}-1)\) when \(p < \frac{2}{\alpha}+\pi\). Using a vaguelette decomposition of fractional Gaussian noise we draw a parallel with certain inverse problems where similar rate results occur.

Keywords: Adaptation, correlated data, deconvolution, degree of ill posedness, fractional Brownian Motion, fractional differentiation, fractional integration, inverse problems, linear processes, long range dependence, L^p loss, nonparametric regression, maxisets, Meyer wavelet, vaguelettes, WaveD.

AMS Subject Classification: Primary 62G05; secondary 62G08, 62G20.