Lp-wavelet regression with correlated errors and inverse problems

Rafal Kułik and Marc Raimondo


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, Lp loss, nonparametric regression, maxisets, Meyer wavelet, vaguelettes, WaveD.

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

This paper is available as a pdf (220kB) file.

Tuesday, July 29, 2008