A SMEARY CENTRAL LIMIT THEOREM FOR MANIFOLDS WITH APPLICATION TO HIGH-DIMENSIONAL SPHERES

• 成果类型: Article
• 署名作者: Eltzner, Benjamin; Huckemann, Stephan F.
• 署名单位: University of Gottingen
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/18-AOS1781
• 发表日期: 2019
• 页码: 3360-3381
• 关键词: extrinsic sample means; center-of-mass; RIEMANNIAN-MANIFOLDS; frechet means; uniqueness; locus; statistics; inference; shapes

摘要

The (CLT) central limit theorems for generalized Frechet means (data descriptors assuming values in manifolds, such as intrinsic means, geodesics, etc.) on manifolds from the literature are only valid if a certain empirical process of Hessians of the Frechet function converges suitably, as in the proof of the prototypical BP-CLT [Ann. Statist. 33 (2005) 1225-1259]. This is not valid in many realistic scenarios and we provide for a new very general CLT. In particular, this includes scenarios where, in a suitable chart, the sample mean fluctuates asymptotically at a scale n(alpha) with exponents alpha < 1/2 with a nonnormal distribution. As the BP-CLT yields only fluctuations that are, rescaled with n(1/2), asymptotically normal, just as the classical CLT for random vectors, these lower rates, somewhat loosely called smeariness, had to date been observed only on the circle. We make the concept of smeariness on manifolds precise, give an example for two-smeariness on spheres of arbitrary dimension, and show that smeariness, although almost never occurring, may have serious statistical implications on a continuum of sample scenarios nearby. In fact, this effect increases with dimension, striking in particular in high dimension low sample size scenarios.