Adaptive estimation of linear functionals by model selection

TL;DR

This paper introduces an adaptive estimation method for linear functionals using Gaussian model selection, providing non-asymptotic risk bounds and demonstrating optimal rates over Besov spaces, with practical simulations.

Contribution

It develops a new Gaussian model selection-based estimator that is adaptive and achieves minimax rates for various functionals, including derivatives and mean estimation.

Findings

The estimator is adaptive and non-asymptotically optimal.

It achieves minimax rates over Besov balls.

Simulations confirm good practical performance.

Abstract

We propose an estimation procedure for linear functionals based on Gaussian model selection techniques. We show that the procedure is adaptive, and we give a non asymptotic oracle inequality for the risk of the selected estimator with respect to the $\mathbb{L}_p$ loss. An application to the problem of estimating a signal or its $r^{th}$ derivative at a given point is developed and minimax rates are proved to hold uniformly over Besov balls. We also apply our non asymptotic oracle inequality to the estimation of the mean of the signal on an interval with length depending on the noise level. Simulations are included to illustrate the performances of the procedure for the estimation of a function at a given point. Our method provides a pointwise adaptive estimator.