High-dimensional simultaneous inference with the bootstrap

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 , s_0 = o(n^1/2/{log (p) log (|G|)^1/2}) and log (|G|) = o(n^1/7) , with p the number of variables, n the sample size and s_0 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 ).