Nonparametric Testing: Efficiency and Distribution-freeness via Optimal Transportation

统计假设检验是统计学家用来在关于潜在数据生成机制的两个相互竞争的假设之间做出决定的正式设置。近年来，非参数方法在统计学理论和实践中变得越来越流行，主要是因为它们提供了比参数模型更大的灵活性。本研究项目研究多维数据的非参数假设检验中的一些问题。现代的计算能力，以及现代科学设备产生的扩大的数据集，大大扩大了这种灵活的统计推断程序的范围。研究人员将开发一个用于多变量数据的“无分布”推断的框架，该框架概括了许多用于分析单变量数据的著名和流行的统计学思想。在合作方面，研究人员将继续进行天文学的跨学科研究。此外，这些研究问题中的一些将构成哥伦比亚大学一名在读博士生的论文。研究人员还计划继续指导本科生暑期实习生的传统。这项研究的主要目的是基于最优传输理论，研究多变量和希尔伯特空间值数据的无分布方法。最优传输理论是最近在应用数学/概率/机器学习中受到广泛关注的数学分支。这些方法将经典的单变量秩基方法推广到多变量数据。在提案的第二部分，研究者将研究非参数检验的渐近相对效率(ARE)，并在零假设下给出当基本检验统计量弱收敛于卡方分布的无限混合时的渐近相对效率(ARE)的特征。这个框架包含了许多在实践中出现的有趣的例子，包括双样本测试、独立性测试、多变量对称性测试、方向性数据推断等。研究人员还将开发一个理论框架来估计在这种情况下的ARE。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.