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竞争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的理论和实践中越来越受欢迎，主要是因为它们比参数模型提供了更大的灵活性。本文研究多维数据的非参数假设检验问题。现代计算能力和现代科学设备产生的扩大的数据集大大增加了这种灵活的统计推断程序的范围。研究人员将开发一个框架，用于“无分布”的推断与多变量数据，概括了许多众所周知的和流行的统计思想用于分析单变量数据。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形成哥伦比亚一名博士生的博士论文。研究者还计划继续指导本科生暑期实习生的传统。本研究的主要目的是研究基于最优传输理论的多变量和希尔伯特空间值数据的无分布方法。最优传输理论是数学的一个分支，最近在应用数学/概率/机器学习中受到了广泛关注。这些方法将经典的基于单变量秩的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，当潜在的检验统计量弱收敛到卡方分布的无限混合时，提供ARE的表征。这个框架包含了许多在实践中出现的有趣的例子，包括双样本检验，独立性检验，多元对称性检验，方向数据的推断，等。研究人员还将制定一个理论框架来评估这种环境中的ARE。该奖项反映了NSF的法定使命，并且通过使用基金会的知识价值和更广泛的影响审查进行评估，被认为值得支持的搜索.

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.