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• W2784537310 abstract "The (CLT) central limit theorems for generalized Frechet means (data descriptors assuming values in stratified spaces, 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 (Bhattacharya and Patrangenaru (2005)). 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 non-normal 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 (Hotz and Huckemann (2015)). 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." @default.
• W2784537310 created "2018-02-02" @default.
• W2784537310 creator A5019677811 @default.
• W2784537310 creator A5049480942 @default.
• W2784537310 date "2018-01-19" @default.
• W2784537310 modified "2023-09-27" @default.
• W2784537310 title "A Smeary Central Limit Theorem for Manifolds with Application to High Dimensional Spheres" @default.
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