COE-F-73
Stein Phenomenon in Estimation of Means Restricted to a Polyhedral Convex Cone
Hisayuki Tsukuma and Tatsuya Kubokawa
University of Tokyo
Abstract
This paper treats the problem of estimating the restricted means of normal distributions
with a known variance, where the means are restricted to a polyhedral convex cone which
includes various restrictions such as positive orthant, simple order, tree order and
umbrella order restrictions. In the context of the simultaneous estimation of the restricted
means, it is of great interest to investigate decision-theoretic properties of the
generalized Bayes estimator against the uniform prior distribution over the polyhedral
convex cone. In this paper, the generalized Bayes estimator is shown to be minimax. It is
also proved that it is admissible in the one- or two-dimensional case, but is improved on by
a shrinkage estimator in the three or more dimensional case. This means that the so-called
Stein phenomenon on the minimax generalized Bayes estimator can be extended to the case
where the means are restricted to the polyhedral convex cone. The risk behaviors of the
estimators are investigated through Monte Carlo simulation, and it is revealed that the
shrinkage estimator has a substantial risk gain.