报告题目：Nonparametric Statistical Inference via Metric Distribution Function in Metric Spaces

报告时间：2024年8月3日下午13：30

报告地点：南湖校区老图书馆四楼左侧研究生5-1学习室

主办单位：数学与统计学院/科研处

报告人：潘文亮

报告人简介：潘文亮，现任中国科学院数学与系统科学研究院副研究员及博士生导师，专注于统计学习算法、医学图像数据分析和度量空间的非参数方法等领域研究。在Annals of Statistics、Journal of the American Statistical Association等统计学顶级杂志上发表了20篇以上学术论文，获得2022年教育部高等学校科学研究优秀成果自然科学类二等奖（排名第二）。主持的科研项目涵盖国家自然科学基金委优秀青年基金、面上项目、青年基金等。同时，担任北京生物医学统计与数据管理研究会副理事长，以及中国现场统计研究会统计交叉科学研究分会副秘书长。

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.