Smeary Limit Theorems for Generalized Fréchet Means on Non-Euclidean Spaces

著名的中心极限定理指出，如果二阶矩存在，同分布随机向量在其期望值附近的波动是渐近正态的，如果用样本容量的平方根重新标度。这一基本事实是众多推论统计方法的基础，没有它，应用统计学就很难成立。在现代模式识别、图像处理和计算生物学的应用推动下，统计理论的重点已经转移到非向量值数据。最初，这些是在圆或球体上的随机方向（例如在气象学和天文学），随机旋转（例如，在机器人和生物力学），或随机元素在复杂的投影空间（来自二维构型的统计形状分析）。在千年之交，采用微分几何方法，两个工作组（Hendriks and Landsman（1998）; Bhattacharya and Patrangenaru（2005））成功地证明了局部图中流形上的模拟极限定理，在适当的条件下，部分技术条件。因此，他们提供了，在这些技术条件下，推理方法，也为非欧几里德数据。对于黎曼流形上所谓的内禀Fréchet均值--这些是所谓Fréchet函数的极小化（需要存在二阶矩）--原则上有三个这样的条件：（a）总体均值的唯一性，（B）总体Fréchet均值处总体Fréchet函数的Hessian的满秩，（c）随机序列中索引的经验Fréchet函数的Hessian的经验过程收敛到总体均值。在圆上，与合作研究伙伴T. Hotz（Ilmenau）在初步工作（Hotz和Huckemann（2015））中，申请人给出了在条件（a）下条件（B）和（c）失败的示例。因此，与中心极限定理相比，渐近速度降低了（Abbildung 1给出了基本的直觉），给出了“模糊”极限定理。本研究的目的是系统地探索这些新的模糊极限定理，并在这些新的渐近性的基础上，为非欧数据开发新的推断统计方法。

The well known central limit theorems states that the fluctuation around their expected value of identically distributed random vectors is asymptotically normal, if rescaled with the squared root of sample size, if second moments exist. This fundamental fact is the basis of numerous inferential statistical methods, and without it, applied statistics is hardly thinkable. Driven by applications in modern pattern recognition, image processing and computational biology, the focus of statistical theory has shifted to non vector-valued data. Initially, these were random direction on the circle or the sphere (e.g. in meteorology and astronomy), random rotations (e.g. in robotics and biomechanics), or random elements in complex projective spaces (from statistical shape analysis of two-dimensional configurations).Around the turn of the millennium, employing differential geometry methods, two workings groups (Hendriks and Landsman (1998); Bhattacharya and Patrangenaru (2005)) succeeded in proving analog limit theorems on manifolds in local charts, under suitable, partially rather technical conditions. Thus, they provided, given these technical conditions, inferential methods, also for non-Euclidean data. For so-called intrinsic Fréchet means on Riemannian manifolds - these are minimizers of so-called Fréchet functions (which require existence of second moments) - there are, in principle, three such conditions:(a) uniqueness of the population mean,(b) full rank of the Hessian of the population Fréchet function at the population Fréchet mean,(c) convergence of the empirical process of the Hessian of the empirical Fréchet function indexed in a random sequence converging to the population mean.On the circle, jointly with the collaborate research partner T. Hotz (Ilmenau), in preliminary work (Hotz and Huckemann (2015)), the applicant gave examples in which under condition (a), conditions (b) and (c) fail. In consequence, in comparison to the central limit theorem, the asymptotic rate is lowered (Abbildung 1 gives the underlying intuition), giving "smeary" limit theorems. It is the aim of this research proposal, to systematically explore these novel smeary limit theorems, and building on these novel asymptotics, develop new inferential statistical methods for non-Euclidean data.