Hermite regression estimation in noisy convolution model

Abstract

In this paper, we consider the following regression model: $y(kT/n)=f\star g(kT/n)+{\varepsilon }_k,k=-n,\dots ,n-1$, $T$ fixed, where $g$ is known and $f$ is the unknown function to be estimated. The errors ${\left({\varepsilon }_k\right)}_{-n\le k\le n-1}$ are independent and identically distributed centered with finite known variance. Two adaptive estimation methods for $f$ are considered by exploiting the properties of the Hermite basis. We study the quadratic risk of each estimator. If $f$ belongs to Sobolev regularity spaces, we derive rates of convergence. Adaptive procedures to select the relevant parameter inspired by the Goldenshluger and Lepski method are proposed and we prove that the resulting estimators satisfy oracle inequalities for sub-Gaussian $\varepsilon$’s. Finally, we illustrate numerically these approaches.