Geometry of Sample Spaces

TL;DR

This paper develops a geometric framework for understanding sample spaces in statistics, especially focusing on the structure of empirical and population means in complex spaces like manifolds and stratified spaces, with applications to Wasserstein spaces.

Contribution

It introduces a geometric perspective on sample spaces using quotient spaces and orbit types, describing their structure and properties in various metric spaces, including Wasserstein space.

Findings

Describes the orbifold and path-metric structure of sample spaces.

Shows the equivalence of the infinite sample space with Wasserstein space.

Establishes asymptotic properties of polymeans, including consistency and normality.

Abstract

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 $M^n$ 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 Fr\'echet 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.