### 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) |
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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