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 $s_0$ denoting the sparsity. The theory is complemented by many empirical results. Our proposed procedures are implemented in the R-package hdi.

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