Omnibus clts for frÉchet means and nonparametric inference on non-euclidean spaces

Research output: Contribution to journalArticle

8 Citations (Scopus)

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 Sd, d > 2, which are also of considerable importance in applications to axial spaces and to landmarksbased image analysis, as these spaces are quotients of spheres under a Lie group G of isometries of Sd.

Original languageEnglish (US)
Pages (from-to)413-428
Number of pages16
JournalProceedings of the American Mathematical Society
Volume145
Issue number1
DOIs
StatePublished - 2017

Fingerprint

Nonparametric Inference
Lie groups
Theorem
Image analysis
Enigma
Stratified Space
Phylogenetic Tree
Sample mean
Image Analysis
Isometry
Central limit theorem
Immediately
Riemannian Manifold
Quotient
Restriction
Methodology

Keywords

  • Fréchet means
  • Inference on manifolds
  • Omnibus central limit theorem
  • Stratified spaces

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Omnibus clts for frÉchet means and nonparametric inference on non-euclidean spaces. / Bhattacharya, Rabindra N; Lin, Lizhen.

In: Proceedings of the American Mathematical Society, Vol. 145, No. 1, 2017, p. 413-428.

Research output: Contribution to journalArticle

@article{697573f8509749fc9c8073aff2951936,
title = "Omnibus clts for fr{\'E}chet means and nonparametric inference on non-euclidean spaces",
abstract = "Two central limit theorems for sample Fr{\'e}chet means are derived, both significant for nonparametric inference on non-Euclidean spaces. The first theorem encompasses and improves upon most earlier CLTs on Fr{\'e}chet 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 Sd, d > 2, which are also of considerable importance in applications to axial spaces and to landmarksbased image analysis, as these spaces are quotients of spheres under a Lie group G of isometries of Sd.",
keywords = "Fr{\'e}chet means, Inference on manifolds, Omnibus central limit theorem, Stratified spaces",
author = "Bhattacharya, {Rabindra N} and Lizhen Lin",
year = "2017",
doi = "10.1090/proc/13216",
language = "English (US)",
volume = "145",
pages = "413--428",
journal = "Proceedings of the American Mathematical Society",
issn = "0002-9939",
publisher = "American Mathematical Society",
number = "1",

}

TY - JOUR

T1 - Omnibus clts for frÉchet means and nonparametric inference on non-euclidean spaces

AU - Bhattacharya, Rabindra N

AU - Lin, Lizhen

PY - 2017

Y1 - 2017

N2 - 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 Sd, d > 2, which are also of considerable importance in applications to axial spaces and to landmarksbased image analysis, as these spaces are quotients of spheres under a Lie group G of isometries of Sd.

AB - 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 Sd, d > 2, which are also of considerable importance in applications to axial spaces and to landmarksbased image analysis, as these spaces are quotients of spheres under a Lie group G of isometries of Sd.

KW - Fréchet means

KW - Inference on manifolds

KW - Omnibus central limit theorem

KW - Stratified spaces

UR - http://www.scopus.com/inward/record.url?scp=84994246352&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84994246352&partnerID=8YFLogxK

U2 - 10.1090/proc/13216

DO - 10.1090/proc/13216

M3 - Article

VL - 145

SP - 413

EP - 428

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 1

ER -