A maxiset approach of a Gaussian noise model

Abstract

We consider the problem of estimating an unknown function $f$ in a homoscedastic Gaussian white noise setting under $\mathbb{L}^p$ risk. The particularity of this model is that it has an intermediate function, say $v$, which complicates the estimate significantly. While varying the assumptions on $v$, we investigate the minimax rate of convergence over two balls of spaces which belong to family of Besov classes. One is defined as usual and the other called 'weighted Besov balls' used $v$ explicitly. Adopting the maxiset approach, we develop a natural hard thresholding procedure which attained the minimax rate of convergence within a logarithmic factor over these weighted balls