A Review of Brown 1971 (in)admissibility results under scale mixtures of Gaussian priors

TL;DR

This paper revisits Brown's 1971 admissibility results for multivariate normal mean estimation, providing a more direct proof for generalized Bayes estimators with scale mixture priors, on its 50th anniversary.

Contribution

It offers an alternative, more direct proof of Brown's admissibility results for a subclass of scale mixture priors, expanding understanding of Bayesian estimators.

Findings

New proof simplifies Brown's original admissibility results.

Extends results to a broader class of scale mixture priors.

Reinforces the significance of Brown's 1971 findings in modern statistical theory.

Abstract

Brown's 1971 paper "Admissible estimators, recurrent diffusions and insoluble boundary value problems" is a landmark in the admissibility literature. It nearly completely settles the issue of admissibility/inadmissibility for estimating the mean of a multivariate normal distribution with identity covariance under sum of squared error loss. We revisit this wonderful tour de force on its 50th anniversary and present an alternative and more direct proof of the result for generalized Bayes estimators corresponding to priors which are a subclass of scale mixtures of spherical normals.