Statistica Sinica 19 (2009), 1479-1489

$\mbox{\boldmath $L^p$}$-WAVELET REGRESSION WITH CORRELATED

ERRORS AND INVERSE PROBLEMS

Rafa Kulik and Marc Raimondo

University of Ottawa and University of Sydney

Abstract: We investigate the global performances of non-linear wavelet estimation in regression models with correlated errors. Convergence properties are studied over a wide range of Besov classes ${\cal B}_{\pi,r}^{s}$ and for a variety of $L^p$ error measures. We consider error distributions with Long-Range-Dependence parameter $\alpha, 0<\alpha\leq 1$. 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 $s= ({\alpha}/{2})({p}/{\pi}-1)$ when $p>{2}/{\alpha}+\pi$. Using a vaguelette decomposition of fractional Gaussian noise we draw a parallel with certain inverse problems where similar rate results occur.

Key words and phrases: 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, maxisets, Meyer wavelet, nonparametric regression, vaguelettes, WaveD.