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

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.