*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.