Bootstrap Methods in High Dimensions: Complex Dependence Structures and Refinements

Bootstrap方法是统计分析中不确定性量化的一类强有力的统计方法。最近的发展表明，引导程序是有效的，在发展理论上有效的推理方法，高维或大规模的数据。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、经济学中常见的多路聚类数据、环境数据和高频金融数据。在这个项目中，PI的目标是为具有复杂依赖结构的大规模数据开发bootstrap方法，并对独立数据的现有bootstrap方法进行改进。此外，该项目将为研究生提供研究培训机会。该项目的第一个研究方向是开发新的自举方法，适用于i）可交换阵列，ii）不规则空间数据，iii）扩散（或更一般的马尔可夫）过程。这些新的Bootstrap方法直接应用于高维数据的推断，如同时和多重检验，以及非参数推断，如同时置信带的构建。第二条研究路线是对独立数据的高维自助方法进行改进。一个是开发的一般交换加权自举在高维制度的高斯乘数自举的误差界。另一个是发展的误差范围的非参数引导，可以解释上级性能的数值实验中增加的dimensionset.This奖项反映了NSF的法定使命，并已被认为是值得支持的，通过使用基金会的智力价值和更广泛的影响审查标准进行评估。

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.