THE STEIN EFFECT FOR FRECHET MEANS

• 成果类型: Article
• 署名作者: Mccormack, Andrew; Hoff, Peter
• 署名单位: Duke University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/22-AOS2245
• 发表日期: 2022
• 页码: 3647-3676
• 关键词: symmetric distributions; shrinkage estimation; EMPIRICAL BAYES; covariance; estimators; improvement; statistics; regression; MANIFOLDS; geometry

摘要

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 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.