Equivariant Estimation of Fréchet Means

Abstract

The Fréchet mean generalizes the concept of a mean to a metric space setting. In this work we consider equivariant estimation of Fréchet means for parametric models on metric spaces that are Riemannian manifolds. The geometry and symmetry of such a space is partially encoded by its isometry group of distance preserving transformations. Estimators that are equivariant under the isometry group take into account the symmetry of the metric space. For some models there exists an optimal equivariant estimator, which necessarily will perform as well or better than other common equivariant estimators, such as the maximum likelihood estimator or the sample Fréchet mean. We derive the general form of this minimum risk equivariant estimator and in a few cases provide explicit expressions for it. A result for finding the Fréchet mean for distributions with radially decreasing densities is presented and used to find expressions for the minimum risk equivariant estimator. In some models the isometry group is not large enough relative to the parametric family of distributions for there to exist a minimum risk equivariant estimator. In such cases, we introduce an adaptive equivariant estimator that uses the data to select a submodel for which there is a minimum risk equivariant estimator. Simulation results show that the adaptive equivariant estimator performs favourably relative to alternative estimators.