TITLE: Simultaneous Bootstrap Inference with High-Dimensional Data

ABSTRACT: We propose bootstrap methodologies for simultaneous inference of low-dimensional parameters with high dimensional data. We focus on simultaneous confidence intervals for individual coefficients in linear regression, although our approach is applicable in much broader contexts. The problem can be solved by de-biasing regularized estimators such as the Lasso. However, the Bonferroni adjustment is overly conservative and asymptotic theory is complicated, especially for non-Gaussian and heteroscedastic errors. We propose residual, wild and empirical bootstrap methodologies for more accurate and robust simultaneous inference and study sample size requirements and other properties of such procedures. Our theory is complemented by many empirical results.