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 $$p \gg n$$ , $$s_0 = o(n^{1/2}/\{\log (p) \log (|G|)^{1/2}\})$$ 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).