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

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 $f(x,u)={\eta }^{(p+n)\mathrm{/}2}f(\eta \{\mathrm{\parallel }x-\theta {\mathrm{\parallel }}^2+\mathrm{\parallel }u{\mathrm{\parallel }}^2\})$, where $\eta$ is unknown. We show that the natural estimator $x$ is admissible for $p=1,2$. Also, for $p\ge 3$, we find classes of generalized Bayes estimators that are admissible within the class of equivariant estimators of the form $\{1-\xi (x\mathrm{/}\mathrm{\parallel }u\mathrm{\parallel })\}x$. In the Gaussian case, a variant of the James–Stein estimator, $[1-\{(p-2)\mathrm{/}(n+2)\}\mathrm{/}\{\mathrm{\parallel }x{\mathrm{\parallel }}^2\mathrm{/}\mathrm{\parallel }u{\mathrm{\parallel }}^2+(p-2)\mathrm{/}(n+2)+1\}]x$, which dominates the natural estimator $x$, is also admissible within this class. We also study the related regression model.