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

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 $f(x,u)={\eta }^{(p+n)/2}f(\eta \{\Vert x-\theta {\Vert }^2+\Vert u{\Vert }^2\})$, 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 $\{1-\xi (x/\Vert u\Vert )\}x$. In the Gaussian case, a variant of the James--Stein estimator, $[1-\{(p-2)/(n+2)\}/\{\Vert x{\Vert }^2/\Vert u{\Vert }^2+(p-2)/(n+2)+1\}]x$, which dominates the natural estimator $x$, is also admissible within this class. We also study the related regression model.

57 pages

References in corpus (1)

• A new class of generalized Bayes minimax ridge regression estimators