Bootstrap Methods in Modern Settings: Inference and Computation

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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.