Inference for Dispersion and Curvature of Random Objects

There are many open questions pertaining to the statistical analysis of random objects, which are increasingly encountered. A major challenge is the absence of linear operations in such spaces. A basic statistical task is to quantify statistical dispersion or spread. For two measures of dispersion for data objects in geodesic metric spaces, Fréchet variance and metric variance, we derive a central limit theorem (CLT) for their joint distribution. This analysis reveals that the Alexandrov curvature of the geodesic space determines the relationship between these two dispersion measures. This suggests a novel test for inferring the curvature of a space based on the asymptotic distribution of the dispersion measures. We demonstrate how this test can be employed to detect the intrinsic curvature of an unknown underlying space, which emerges as a joint property of the space and the underlying probability measure that generates the random objects. We investigate the asymptotic properties of the test and its finite-sample behavior for various data types, including distributional data and point cloud data. We illustrate the proposed inference for intrinsic curvature of random objects using gait synchronization data represented as symmetric positive definite matrices and energy compositional data on the sphere. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

随机对象的统计分析正愈发常见，相关研究仍存在大量未决问题。此类空间的核心挑战之一在于缺乏线性运算。针对测地度量空间中数据对象的两种离散度度量——弗雷歇方差（Fréchet variance）与度量方差（metric variance），我们推导了二者联合分布的中心极限定理（CLT）。该分析表明，测地空间的亚历山德罗夫曲率（Alexandrov curvature）决定了这两种离散度度量之间的关联关系。这一结论提出了一种基于离散度度量的渐近分布推断空间曲率的全新检验方法。我们展示了如何利用该检验方法检测未知底层空间的内在曲率，而该曲率实则为空间与生成随机对象的底层概率测度的联合属性。我们针对多种数据类型（包括分布数据与点云数据），研究了该检验的渐近性质与有限样本表现。我们以表示为对称正定矩阵（symmetric positive definite matrices）的步态同步数据，以及球面上的能量成分数据为例，阐释了针对随机对象内在曲率的所提推断方法。本文的补充材料可在线获取，其中包含可用于复现研究成果的材料的标准化说明。