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

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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.