A maxiset approach of a Gaussian white noise model

Abstract

This paper is devoted to the estimation of an unknown function $f$ in the framework of a Gaussian white noise model. The noise process is represented by $t\rightarrow\frac{1}{\sqrt{n}}\int_{0}^{t}g(x) dB_x$, where the variance function $g$ is assumed to be known. Adopting the maxiset point of view, we study the performance of two different hard thresholding estimators in $\mathbb{L}^p$ norm. In a first part, we expand $f$ on a compactly supported wavelet basis $\{\psi_{\lambda}(.); \ \lambda\in\Lambda\}$. From this decomposition, we use some results about the heteroscedastic white noise model to construct a well adapted hard thresholding estimator and to exhibit the associated maxiset. In a second part, we introduce the classes of Muckenhoupt weights and we use this analytical tools to investigate the geometrical properties of warped wavelet basis $\{\psi_{\lambda}(T(.)); \ \lambda\in\Lambda\}$ in $\mathbb{L}^p$ norm. Expanding $f$ on such a basis and considering the associated hard thresholding estimator, we investigate the maxiset properties under some assumptions on $g$. We finally apply this result to find an upper bound over weighted Besov spaces