High-dimensional simultaneous inference with the bootstrap

Abstract

We propose a residual and wild bootstrap methodology for individual and simultaneous inference in high-dimensional linear models with possibly non-Gaussian and heteroscedastic errors. We establish asymptotic consistency for simultaneous inference for parameters in groups G, where p≫n, s0=o(n1/2/{log(p)log(|G|)1/2}) and log(|G|)=o(n1/7), with p the number of variables, n the sample size and s0 the sparsity. The theory is complemented by many empirical results. Our proposed procedures are implemented in the R-package hdi (Meier et al. hdi: high-dimensional inference. R package version 0.1-6, 2016).