A sharp boundary for SURE-based admissibility for the Normal means problem under unknown scale

TL;DR

This paper establishes a precise boundary for the admissibility of Stein-type estimators in the normal means problem with unknown scale, linking it to the James-Stein estimator and extending to spherical distributions.

Contribution

It introduces a sharp boundary criterion for quasi-admissibility of Stein-type estimators under unknown scale, connecting it to known variance cases and prior classes.

Findings

Identifies a sharp boundary between quasi-admissible and quasi-inadmissible estimators.

Links the boundary to the optimal James-Stein estimator.

Extends analysis to spherically symmetric distributions.

Abstract

We consider quasi-admissibility/inadmissibility of Stein-type shrinkage estimators of the mean of a multivariate normal distribution with covariance matrix an unknown multiple of the identity. Quasi-admissibility/inadmissibility is defined in terms of non-existence/existence of a solution to a differential inequality based on Stein's unbiased risk estimate (SURE). We find a sharp boundary between quasi-admissible and quasi-inadmissible estimators related to the optimal James-Stein estimator. We also find a class of priors related to the Strawderman class in the known variance case where the boundary between quasi-admissibility and quasi-inadmissibility corresponds to the boundary between admissibility and inadmissibility in the known variance case. Additionally, we also briefly consider generalization to the case of general spherically symmetric distributions with a residual vector.