  The Frechet mean is a useful description of location for a probability
distribution on a metric space that is not necessarily a vector space. This
article considers simultaneous estimation of multiple Frechet means from a
decision-theoretic perspective, and in particular, the extent to which the
unbiased estimator of a Frechet mean can be dominated by a generalization of
the James-Stein shrinkage estimator. It is shown that if the metric space
satisfies a non-positive curvature condition, then this generalized James-Stein
estimator asymptotically dominates the unbiased estimator as the dimension of
the space grows. These results hold for a large class of distributions on a
variety of spaces - including Hilbert spaces - and therefore partially extend
known results on the applicability of the James-Stein estimator to non-normal
distributions on Euclidean spaces. Simulation studies on metric trees and
symmetric-positive-definite matrices are presented, numerically demonstrating
the efficacy of this generalized James-Stein estimator.
