Admissible Bayes equivariant estimation of location vectors for spherically symmetric distributions with unknown scale

TL;DR

This paper explores the admissibility of estimators for the mean vector in spherically symmetric distributions with unknown scale, proposing generalized Bayes estimators and a variant of James--Stein estimator that outperform the natural estimator.

Contribution

It introduces new admissible estimators for the mean vector under unknown scale in spherical distributions, extending classical results and including a novel James--Stein type estimator.

Findings

The natural estimator is admissible for p=1,2.

For p≥3, certain generalized Bayes estimators are admissible.

A James--Stein type estimator dominates the natural estimator in the Gaussian case.

Abstract

This paper investigates estimation of the mean vector under invariant quadratic loss for a spherically symmetric location family with a residual vector with density of the form $ f(x,u)=\eta^{(p+n)/2}f(\eta\{\|x-\theta\|^2+\|u\|^2\}) $, where $\eta$ is unknown. We show that the natural estimator $x$ is admissible for $p=1,2$. Also, for $p\geq 3$, we find classes of generalized Bayes estimators that are admissible within the class of equivariant estimators of the form $\{1-\xi(x/\|u\|)\}x$. In the Gaussian case, a variant of the James--Stein estimator, $[1-\{(p-2)/(n+2)\}/\{\|x\|^2/\|u\|^2+(p-2)/(n+2)+1\}]x$, which dominates the natural estimator $x$, is also admissible within this class. We also study the related regression model.