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

2017/10/08 by Yuzo Maruyama, Maruyama, Yuzo, William E. Strawderman +1
Mathematics · #62C15 #FOS: Mathematics #Statistics Theory (math.ST) #math.ST #msc:62C15 #stat.TH

paper · pdf · doi:10.48550/arxiv.1710.02794

57 pages

arxiv created 2017/10/08 · arxiv updated 2017/10/10

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)=η^(p+n)/2f(η\‖x-θ‖²+‖u‖²\) , where η is unknown. We show that the natural estimator x is admissible for p=1,2. Also, for p≥ 3, we find classes of generalized Bayes estimators that are admissible within the class of equivariant estimators of the form \1-ξ(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.

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