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

: Primary 62G05; secondary 62G08, 62G20.

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 Tuesday, July 29, 2008