Nonparametric Statistical Inference via Metric Distribution Function in Metric Spaces

• 成果类型: Article
• 署名作者: Wang, Xueqin; Zhu, Jin; Pan, Wenliang; Zhu, Junhao; Zhang, Heping
• 署名单位: Chinese Academy of Sciences; University of Science & Technology of China, CAS; Sun Yat Sen University; University of London; London School Economics & Political Science; Chinese Academy of Sciences; Yale University; University of Toronto
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459
• DOI: 10.1080/01621459.2023.2277417
• 发表日期: 2024
• 页码: 2772-2784
• 关键词: frechet regression; dependence; tests

摘要

The distribution function is essential in statistical inference and connected with samples to form a directed closed loop by the correspondence theorem in measure theory and the Glivenko-Cantelli and Donsker properties. This connection creates a paradigm for statistical inference. However, existing distribution functions are defined in Euclidean spaces and are no longer convenient to use in rapidly evolving data objects of complex nature. It is imperative to develop the concept of the distribution function in a more general space to meet emerging needs. Note that the linearity allows us to use hypercubes to define the distribution function in a Euclidean space. Still, without the linearity in a metric space, we must work with the metric to investigate the probability measure. We introduce a class of metric distribution functions through the metric only. We overcome this challenging step by proving the correspondence theorem and the Glivenko-Cantelli theorem for metric distribution functions in metric spaces, laying the foundation for conducting rational statistical inference for metric space-valued data. Then, we develop a homogeneity test and a mutual independence test for non-Euclidean random objects and present comprehensive empirical evidence to support the performance of our proposed methods. Supplementary materials for this article are available online.