Modeling observations as random distributions embedded within Wasserstein spaces is becoming increasingly popular across scientific fields, as it captures the variability and geometric structure of the data more effectively. However, the distinct geometry and unique properties of Wasserstein spaces pose challenges to the application of conventional statistical tools, which are primarily designed for Euclidean spaces. Consequently, adapting and developing new methodologies for analysis within Wasserstein spaces has become essential. The space of distributions on $\mathbb{R}^d$ with $d>1$ is not linear, and "mimic" the geometry of a Riemannian manifold. In this paper, we extend the concept of statistical depth to distribution-valued data, introducing the notion of Wasserstein spatial depth. This new measure provides a way to rank and order distributions, enabling the development of order-based clustering techniques and inferential tools. We show that Wasserstein spatial depth (WSD) preserves critical properties of conventional statistical depths, notably, ranging within $[0,1]$, transformation and geodesic invariance, vanishing at infinity, reaching a maximum at the geometric median, and continuity. Regarding robustness, we characterize the breakdown points of the empirical depth regions and the influence function of the WSD. Additionally, the population WSD has a straightforward plug-in estimator based on sampled empirical distributions. We establish the estimator's consistency and asymptotic normality. We also provide a two-sample test for populations of distributions based on the WSD. Finally, extensive simulations and a real-data application showcase the practical efficacy of the WSD.

翻译：将观测数据建模为嵌入Wasserstein空间中的随机分布在科学领域日益普及，该方法能更有效地捕捉数据的变异性和几何结构。然而，Wasserstein空间独特的几何性质和独特属性，给传统主要面向欧氏空间设计的统计工具的应用带来了挑战。因此，针对Wasserstein空间内的分析进行方法调适与开发至关重要。当$d>1$时，$\mathbb{R}^d$上分布的集合并非线性空间，而是"模拟"了黎曼流形的几何特征。本文我们将统计深度的概念扩展到分布值数据，提出了Wasserstein空间深度这一概念。这一新指标提供了一种对分布进行排序和排序的方法，能支持基于排序的聚类技术和推断工具的开发。我们证明了Wasserstein空间深度保留了传统统计深度的关键性质，特别是其取值范围为$[0,1]$，具备变换不变性和测地线不变性，在无穷远处趋于零，在几何中位数处达到最大值，且具有连续性。在稳健性方面，我们刻画了经验深度区域的崩溃点以及Wasserstein空间深度的影响函数。此外，总体Wasserstein空间深度存在一个基于抽样经验分布的简便插值估计量。我们建立了该估计量的一致性和渐近正态性。同时，基于Wasserstein空间深度，我们还为分布总体提供了一个双样本检验。最后，广泛的模拟实验和一项实际数据应用展示了Wasserstein空间深度实际应用中的有效性。