The Stein hull

TL;DR

This paper explores the use of risk hulls in statistical linear inverse problems, analyzing threshold estimators and proposing a penalty for adaptive estimation with improved accuracy.

Contribution

It introduces a novel perspective linking threshold estimators to risk hulls, and proposes a near-optimal penalty for adaptive linear inverse problem estimation.

Findings

Threshold estimators can be viewed as risk hull minimizers.

A well-chosen penalty improves estimator accuracy.

Proposed penalty is close to the theoretical lower bound.

Abstract

We are interested in the statistical linear inverse problem $Y=Af+\epsilon\xi$, where $A$ denotes a compact operator and $\epsilon\xi$ a stochastic noise. In a first time, we investigate the link between some threshold estimators and the risk hull point of view introduced in (5). The penalized blockwise Stein's rule plays a central role in this study. In particular, this estimator may be considered as a risk hull minimization method, provided the penalty is well-chosen. Using this perspective, we study the properties of the threshold and propose an admissible range for the penalty leading to accurate results. We eventually propose a penalty close to the lower bound of this range. The risk hull point of view provides interesting tools for the construction of adaptive estimators. It sheds light on the processes governing the behavior of linear estimators. The variability of the problem may be indeed quite large and should be carefully controlled.