The Stein effect for Fréchet means

Abstract

The Fréchet 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 Fréchet means from a decision-theoretic perspective, and in particular, the extent to which the unbiased estimator of a Fréchet mean can be dominated by a generalization of the James–Stein shrinkage estimator. It is shown that if the metric space satisfies a nonpositive 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 nonnormal distributions on Euclidean spaces. Simulation studies on phylogenetic trees and symmetric positive definite matrices are presented, numerically demonstrating the efficacy of this generalized James–Stein estimator.