On minimax nonparametric estimation of signal in Gaussian noise

TL;DR

This paper analyzes the asymptotic minimax properties of linear estimators for nonparametric signal estimation in Gaussian noise, revealing that Pinsker estimators have suboptimal convergence rates on certain function spaces.

Contribution

It identifies Sobolev space balls as maxisets for Pinsker estimators and compares their convergence rates to other linear estimators.

Findings

Pinsker estimators are asymptotically minimax on Sobolev balls.

Rates of convergence for Pinsker estimators are worse than those for some linear estimators.

Sobolev spaces are established as maxisets for these estimators.

Abstract

For the problem of nonparametric estimation of signal in Gaussian noise we point out the strong asymptotically minimax estimators on maxisets for linear estimators (see \cite{ker93,rio}). It turns out that the order of rates of convergence of Pinsker estimator on this maxisets is worse than the order of rates of convergence for the class of linear estimators considered on this maxisets. We show that balls in Sobolev spaces are maxisets for Pinsker estimators.