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.

翻译：在统计学中，独立同分布的随机样本不具有自然顺序，其统计量通常对样本顺序的置换保持不变。因此，空间 $M$ 中的 $n$ 个样本可视为商空间 $M^n$ 在置换群作用下的某个元素。本文以样本空间的这一定义及相关的轨道类型概念为出发点，旨在发展统计学的几何视角。我们致力于建立普适的数学框架，以研究从光滑黎曼流形到一般分层空间中经验均值与总体均值的性质。当 $M$ 分别为流形或路径度量空间时，我们完整描述了样本空间的轨道折叠结构与路径度量结构——即便在 $M$ 为欧几里得空间时，这些结论也非平凡。我们证明无穷样本空间在 Gromov-Hausdorff 类型的意义下存在，并恰好与 $M$ 上概率分布的 Wasserstein 空间一致。我们揭示 Fréchet 均值和 $k$-均值可视为 Wasserstein 空间中向 1-骨架或 $k$-骨架的度量投影，并定义了更广义的“多均值”概念。这种通过度量投影的几何刻画同样适用于样本均值与总体均值，我们利用它建立了多均值的渐近性质（如一致性与渐近正态性）。