Geometry of sample spaces

In statistics, independent, identically distributed random samples do not carry a natural ordering, and their statistics are typically invariant with respect to permutations of their order. Thus, an n-sample in a space M can be considered as an element of the quotient space of Mn modulo the permutation group. The present paper takes this definition of sample space and the related concept of orbit types as a starting point for developing a geometric perspective on statistics. We aim at deriving a general mathematical setting for studying the behavior of empirical and population means in spaces ranging from smooth Riemannian manifolds to general stratified spaces.We fully describe the orbifold and path-metric structure of the sample space when M is a manifold or path-metric space, respectively. These results are non-trivial even when M is Euclidean. We show that the infinite sample space exists in a Gromov-Hausdorff type sense and coincides with the Wasserstein space of probability distributions on M. We exhibit Frechet means and k-means as metric projections onto 1-skeleta or k-skeleta in Wasserstein space, and we define a new and more general notion of polymeans. This geometric characterization via metric projections applies equally to sample and population means, and we use it to establish asymptotic properties of polymeans such as consistency and asymptotic normality.& COPY; 2023 Elsevier B.V. All rights reserved.

Automated summary

This paper develops a geometric perspective on statistics by defining the sample space in the context of independent, identically distributed random samples. The orbifold and path-metric structure of the sample space are fully described, and the infinite sample space is shown to coincide with the Wasserstein space of probability distributions. Metric projections onto 1-skeleta or k-skeleta in Wasserstein space are used to characterize Frechet means and k-means, and a new notion of polymeans is introduced.