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Omnibus CLTs for Fréchet means and nonparametric inference on non-Euclidean spaces


Authors: Rabi Bhattacharya and Lizhen Lin
Journal: Proc. Amer. Math. Soc. 145 (2017), 413-428
MSC (2010): Primary 60F05, 62E20, 60E05, 62G20
DOI: https://doi.org/10.1090/proc/13216
Published electronically: July 26, 2016
MathSciNet review: 3565392
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Abstract: Two central limit theorems for sample Fréchet means are derived, both significant for nonparametric inference on non-Euclidean spaces. The first theorem encompasses and improves upon most earlier CLTs on Fréchet means and broadens the scope of the methodology beyond manifolds to diverse new non-Euclidean data, including those on certain stratified spaces which are important in the study of phylogenetic trees. It does not require that the underlying distribution $ Q$ have a density and applies to both intrinsic and extrinsic analysis. The second theorem focuses on intrinsic means on Riemannian manifolds of dimensions $ d>2$ and breaks new ground by providing a broad CLT without any of the earlier restrictive support assumptions. It makes the statistically reasonable assumption of a somewhat smooth density of $ Q$. The excluded case of dimension $ d=2 $ proves to be an enigma, although the first theorem does provide a CLT in this case as well under a support restriction. The second theorem immediately applies to spheres $ S^d$, $ d>2$, which are also of considerable importance in applications to axial spaces and to landmarks-based image analysis, as these spaces are quotients of spheres under a Lie group $ \mathcal G $ of isometries of $ S^d$.


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Additional Information

Rabi Bhattacharya
Affiliation: Department of Mathematics, The University of Arizona, Tucson, Arizona 85721
Email: rabi@math.arizona.edu

Lizhen Lin
Affiliation: Department of Statistics and Data Sciences, The University of Texas at Austin, Austin, Texas 78712
Email: lizhen.lin@austin.utexas.edu

DOI: https://doi.org/10.1090/proc/13216
Keywords: Inference on manifolds, Fr\'echet means, Omnibus central limit theorem, stratified spaces
Received by editor(s): February 28, 2015
Received by editor(s) in revised form: November 12, 2015, and March 27, 2016
Published electronically: July 26, 2016
Communicated by: Mark M. Meerschaert
Article copyright: © Copyright 2016 American Mathematical Society