Bootstrap Methods in Modern Settings: Inference and Computation

Bootstrap方法在统计方法学中起着核心作用，并且是最广泛使用的不确定性量化工具之一。虽然自举方法有广泛的文献，但其适用范围在现代统计问题中还远未得到充分理解。当数据由具有许多未知参数的高维模型描述时，或者当数据量超过计算约束时，情况尤其如此。受这些挑战的激励，所提出的研究有两个主要目标，即（1）分析自举方法在高维模型中的理论有效性，以及（2）开发使用自举方法增强大规模计算的新方法。该研究生将专注于分析高维和大规模数据的bootstrap方法。关于第一个目标，研究将集中在克服非参数自助法的某些局限性，特别是在高维主成分分析的背景下。这将是追求通过一个泛化的参数引导，是专门为感兴趣的统计。此类统计量的示例包括由样本协方差矩阵的特征值产生的统计量。对于第二个目标，本研究将探索自助方法作为一种系统的方法来估计随机算法的错误。鉴于引导方法在历史上被标记为计算密集型，该应用基于相对不同的观点，因为它试图使用引导方法来辅助计算。特别是，自助法将被开发，以获得数值误差估计，提供了一个实际的替代最坏情况下的错误analysis.This奖项反映了NSF的法定使命，并已被认为是值得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的支持。

Bootstrap methods play a central role in statistical methodology, and are among the most widely used tools for uncertainty quantification. Although bootstrap methods have an extensive literature, their scope of applicability is far from being fully understood in modern statistical problems. This is especially the case when data are described by high-dimensional models with many unknown parameters, or when then the amount of data exceeds computational constraints. Motivated by these challenges, the proposed research has two main objectives, which are to (1) analyze the theoretical validity of bootstrap methods in high-dimensional models, and (2) develop novel ways of using bootstrap methods to enhance large-scale computations. The graduate student will focus on the analysis of bootstrap methods for high-dimensional and large-scale data. With regard to the first objective, the research will focus on overcoming certain limitations of the non-parametric bootstrap, particularly in the context of high-dimensional principal components analysis. This will be pursued through a generalization of the parametric bootstrap that is tailored to the statistics of interest. Examples of such statistics include those arising from the eigenvalues of sample covariance matrices. For the second objective, the research will explore bootstrap methods as a systematic way to estimate the errors of randomized algorithms. Given that bootstrap methods have been historically labeled as computationally intensive, this application is based on a relatively distinct perspective, since it seeks to use bootstrap methods to assist computation. In particular, bootstrap methods will be developed to obtain numerical error estimates that offer a practical alternative to worst-case error analysis.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.