Admissible Bayes equivariant estimation of location vectors for spherically symmetric distributions with unknown scale Article Swipe
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· 2020
· Open Access
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· DOI: https://doi.org/10.1214/19-aos1837
· OA: W3029919818
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.