Asymptotic data analysis on manifolds

TL;DR

This paper extends the concept of mean and median to distributions on manifolds, analyzing their asymptotic properties and providing inference methods, especially for non-identically distributed samples on submanifolds.

Contribution

It generalizes statistical functionals like mean and median to manifold-valued data and studies their asymptotic behavior, including in multisample scenarios.

Findings

Established convergence properties related to the cutlocus.

Developed asymptotic inference methods for manifold data.

Applied results to multisample, non-i.i.d. data.

Abstract

Given an m-dimensional compact submanifold $\mathbf{M}$ of Euclidean space $\mathbf{R}^s$, the concept of mean location of a distribution, related to mean or expected vector, is generalized to more general $\mathbf{R}^s$-valued functionals including median location, which is derived from the spatial median. The asymptotic statistical inference for general functionals of distributions on such submanifolds is elaborated. Convergence properties are studied in relation to the behavior of the underlying distributions with respect to the cutlocus. An application is given in the context of independent, but not identically distributed, samples, in particular, to a multisample setup.