Huber means on Riemannian manifolds

• 成果类型: Article; Early Access
• 署名作者: Lee, Jongmin; Jung, Sungkyu
• 署名单位: Pusan National University; Seoul National University (SNU); Seoul National University (SNU)
• 刊物名称: JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY
• ISSN/ISSBN: 1369-7412
• DOI: 10.1093/jrsssb/qkaf054
• 发表日期: 2025
• 关键词: robust estimation; sample theory; regression; statistics

摘要

This article introduces Huber means on Riemannian manifolds, providing a robust alternative to the Fr & eacute;chet mean by integrating elements of both L2 and L1 loss functions. The Huber means are designed to be highly resistant to outliers while maintaining efficiency, making it a valuable generalization of Huber's M-estimator for manifold-valued data. We comprehensively investigate the statistical and computational aspects of Huber means, demonstrating their utility in manifold-valued data analysis. Specifically, we establish nearly minimal conditions for ensuring the existence and uniqueness of the Huber mean and discuss regularity conditions for unbiasedness. The Huber means are consistent and enjoy the central limit theorem. Additionally, we propose a novel moment-based estimator for the limiting covariance matrix, which is used to construct a robust one-sample location test procedure and an approximate confidence region for location parameters. The Huber mean is shown to be highly robust and efficient in the presence of outliers or under heavy-tailed distributions. Specifically, it achieves a breakdown point of at least 0.5, the highest among all isometric equivariant estimators, and is more efficient than the Fr & eacute;chet mean under heavy-tailed distributions.