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 $$p \gg n$$ p ≫ n , $$s_0 = o(n^{1/2}/\{\log (p) \log (|G|)^{1/2}\})$$ s 0 = o ( n 1 / 2 / { log ( p ) log ( | G | ) 1 / 2 } ) and $$\log (|G|) = o(n^{1/7})$$ log ( | G | ) = o ( n 1 / 7 ) , with p the number of variables, n the sample size and $$s_0$$ 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).