Bootstrap Methods in High Dimensions: Complex Dependence Structures and Refinements

自举法构成了统计分析中不确定性量化的一类强大的统计方法。最近的发展表明，自举在为高维或大规模数据开发理论上有效的推理方法方面是有效的。现代数据的可用性使我们能够访问具有复杂依赖结构的大规模数据，因此迫切需要开发高质量的推理方法。此类数据的示例包括在网络上观察到的数据、经常在经济学中看到的多路聚类数据、环境数据和高频金融数据。在本项目中，PI的目标是针对这种具有复杂依赖结构的大规模数据开发自举方法，并对现有的独立数据自举方法进行改进。此外，该项目将为研究生提供研究培训机会。这个项目的第一个研究目标是开发新的自举方法，有效地在高维i)可交换阵列，ii)不规则间隔的空间数据，和iii)扩散（或更一般地说，马尔可夫）过程。这些新的自举方法直接应用于高维数据的推理，如同时检验和多重检验，以及非参数推理，如同时置信带的构建。研究的第二步是对具有独立数据的高维自举方法进行改进。一是在高维区域中为一般的可交换加权自举法开发与高斯乘法器自举法相当的误差界。另一个是建立非参数自举的误差范围，以解释在数值实验中在增加维数设置下的优越性能。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Bootstrap methods constitute a class of powerful statistical methodologies for uncertainty quantification in statistical analysis. Recent developments demonstrate that bootstraps are effective in developing theoretically valid inferential methods for high-dimensional or large-scale data. Modern-day data availability allows us to access large-scale data with complex dependence structures, for which there is a serious need to develop high-quality inference methods. Examples of such data include data observed on networks, multiway-clustered data often seen in economics, environmental data, and high-frequency financial data. In this project, the PI aims to develop bootstrap methods for such large-scale data with complex dependence structures, as well as make refinements on existing bootstrap methods for independent data. Additionally, the project will provide research training opportunities for graduate students.The first line of research this project aims at is to develop novel bootstrap methods effective in high dimensions for i) exchangeable arrays, ii) irregularly spaced spatial data, and iii) diffusion (or, more generally, Markov) processes. Those new bootstrap methods have direct applications to inference with high dimensional data such as simultaneous and multiple testing and nonparametric inference such as the construction of simultaneous confidence bands. The second line of research is to make refinements on high dimensional bootstrap methodology with independent data. One is to develop error bounds for the general exchangeably weighted bootstrap in the high dimensional regime that are comparable to the Gaussian multiplier bootstrap. The other is to develop error bounds for the nonparametric bootstrap that can explain the superior performance in numerical experiments in the increasing dimension setup.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.