Nonparametric Testing: Efficiency and Distribution-freeness via Optimal Transportation

非参数测试：通过最佳运输实现效率和无分配

• 批准号: 2311062
• 负责人: Bodhisattva Sen
• 金额: $ 30万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2311062&HistoricalAwards=false
• 关键词: Nonparametric; Testing; Efficiency; Distribution; freeness

Statistical hypothesis testing is the formal setup a statistician employs to decide between two competing hypotheses about the underlying data generating mechanism. Nonparametric methods have become increasingly popular in the theory and practice of statistics in recent times, primarily because of the greater flexibility they offer over parametric models. This research project investigates some problems in nonparametric hypothesis testing for multi-dimensional data. Modern computational capabilities, and the expanded data sets produced by modern scientific equipment have greatly increased the scope of such flexible statistical inference procedures. The investigator will develop a framework for "distribution-free" inference with multivariate data that generalizes many well-known and popular statistical ideas used for analyzing univariate data. On the collaborative front, the investigator will continue interdisciplinary research in astronomy. Further, some of these research problems will form the dissertation thesis of a current PhD student at Columbia. The investigator also plans to continue the tradition of mentoring undergraduate summer interns.The main thrust of this research is to study distribution-free methods for multivariate and Hilbert space-valued data, based on the theory of optimal transport -- a branch of mathematics that has received much attention lately in applied mathematics/probability/machine learning. These methods generalize the classical univariate rank-based methods to multivariate data. In the second part of the proposal, the investigator will study the asymptotic relative efficiency (ARE) of nonparametric tests and provide a characterization of ARE when the underlying test statistics converge weakly to an infinite mixtures of chi-square distributions, under the null hypothesis. This framework includes many interesting examples that arise in practice, including two-sample testing, independence testing, testing multivariate symmetry, inference on directional data, etc. The investigator will also develop a theoretical framework for estimating the ARE in this setting.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

统计假设检验是统计学家采用的正式设置来决定有关基本数据生成机制的两个相互竞争的假设。近来，非参数方法在统计学的理论和实践中变得越来越流行，这主要是因为它们比参数模型具有更大的灵活性。该研究项目研究了多维数据的非参数假设测试中的一些问题。现代计算能力以及现代科学设备生产的扩展数据集大大提高了这种灵活的统计推断程序的范围。研究人员将使用多元数据来开发一个“无分配”推断的框架，该框架概括了用于分析单变量数据的许多知名和流行的统计思想。在协作方面，研究人员将继续在天文学领域进行跨学科研究。此外，其中一些研究问题将构成哥伦比亚当前博士生的论文论文。研究人员还计划继续指导本科暑期实习生的传统。这项研究的主要目的是基于最佳运输理论研究多元和希尔伯特太空值数据的无分配方法，这是数学的一个分支，该分支在应用中在应用数学/概率/机器学习方面引起了很多关注。这些方法将基于经典的单变量方法推广到多元数据。在该提案的第二部分中，研究人员将研究非参数测试的渐近相对效率（IS），并在零假设下，基础测试统计量微弱地汇聚到Chi-square分布的无限混合物时提供的特征为IS。该框架包括许多在实践中出现的有趣示例，包括两样本测试，独立性测试，测试多元对称性，定向数据的推断等。研究者还将开发一个理论框架，以估算此环境。该奖项反映了NSF的法定任务，并通过评估了基金会的范围，并通过评估了基金会的范围。