Metric Distribution Function

：王学钦（中国科学技术大学）
：2022-06-07 15:00
：厦大海韵校区实验楼105报告厅

报告人：王学钦（中国科学技术大学）

时  间：6月7日下午15:00

地  点：厦大海韵校区实验楼105报告厅

内容摘要:

Statistical inference aims to use observed samples to learn the unknown properties of a population. It has become an integral step in scientific reasoning. A building block of nonparametric statistical inference is distribution function. The distribution function and samples are connected to form a directed closed loop by the correspondence theorem in measure theory and the Glivenko-Cantelli and Donsker properties in statistics, and this connection creates a paradigm for statistical inference. However, existing distribution functions are defined in Euclidean spaces. Those distribution functions are no longer convenient to use or applicable in characterizing the rapidly evolving data objects of complex nature. Thus, 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 an Euclidean space, but without the linearity in a metric space, we must work with balls as the basis of the metric topology in defining a probability measure. We introduce a class of novel quasi-distribution functions, or ball functions, for metric space-valued random objects. A ball requires a center and a radius. The center depends on the random point of interest, and the radius is determined by the distance between the center and another random point. Working with balls in defining a probability measure is particularly challenging because unlike hypercubes, the intersection of two balls may not be a ball. We overcome this challenge to prove the correspondence theorem and the Glivenko-Cantelli theorem in metric spaces that lie the foundation for conducting rational statistical inference for metric space-valued data. Based on ball function, we develop statistical methods for homogeneity test, mutual independence test, and hierarchical clustering for non-Euclidean random objects, and present comprehensive empirical evidence to support the performance of our proposed methods.

个人简介：

王学钦，中国科学技术大学教授。2003年博士毕业于纽约州立大学宾厄姆顿分校。2012年入选教育部新世纪优秀人才支持计划学者， 2013年获得国家优秀青年研究基金，2021年入选国家级领军人才。他还担任教育部高等学校统计学类专业教学指导委员会委员、统计学国际期刊《JASA》、《SII》、《JCS》的Associate Editor、高等教育出版社《Lecture Notes: Data Science, Statistics and Probability》系列丛书的副主编、中国现场统计研究会数据科学与人工智能分会副理事长和中国青年统计学家协会副会长等。

 

联系人：周达