非欧几里得空间上Fréchet平均和非参数推断的综合CLT
Omnibus CLTs for Fréchet means and nonparametric inference on non-Euclidean spaces
摘要：
得到了样本Fréchet平均的两个中心极限定理，这两个定理对于非欧几里得空间上的非参数推断都有重要意义.第一个，定理2.2，包括和改进最早的CLTs上Fréchet的手段和扩大范围的方法超越流形，以不同的新的非欧几里德数据，包括那些在某些分层空间是重要的系统发育树的研究。它不要求底层分布$Q$具有密度，并且适用于内在和外在分析。第二个定理，定理3.3，集中在维数为$d>2$的黎曼流形上的内在平均，并通过提供一个广泛的CLT而没有任何早期的限制性支持假设来开辟新的领域。它对$Q$的某种平滑密度做出了统计上合理的假设。维数$d=2$的排除情形被证明是一个谜，尽管第一个定理在这种情形下也在支撑限制下提供了一个CLT。定理3.3直接适用于球体$S^d$、$d>2$，它们在轴向空间和基于界标的图像分析的应用中也是相当重要的，因为这些空间是球体在$S^d$的等距李群$\mathcal G $下的等价物。
英文摘要：
Two central limit theorems for sample Fréchet means are derived, both significant for nonparametric inference on non-Euclidean spaces. The first one, Theorem 2.2, 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, Theorem 3.3, 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. Theorem 3.3 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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