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

The Annals of Statistics · 2020 · Volume 48 · Issue 2

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.

Keywords

62C15, 62J07, Admissibility, Bayes equivariance, generalized Bayes, Stein’s phenomenon

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