### 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 landmarksbased image analysis, as these spaces are quotients of spheres under a Lie group G of isometries of S^{d}.

Original language | English (US) |
---|---|

Pages (from-to) | 413-428 |

Number of pages | 16 |

Journal | Proceedings of the American Mathematical Society |

Volume | 145 |

Issue number | 1 |

DOIs | |

State | Published - 2017 |

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### 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.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 145, no. 1, pp. 413-428. https://doi.org/10.1090/proc/13216

}

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 -