High-dimensional simultaneous inference with the bootstrap

By Ruben Dezeure, Peter Buhlmann, Cun-Hui Zhang. Test (Madrid), 2016

157 citations recorded by the source corpus.

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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \gg n$$\end{document}, s0=o(n1/2/{log(p)log(|G|)1/2})\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0 = o(n^{1/2}/\{\log (p) \log (|G|)^{1/2}\})$$\end{document} and log(|G|)=o(n1/7)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log (|G|) = o(n^{1/7})$$\end{document}, with p the number of variables, n the sample size and s0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0$$\end{document} 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).

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/10701988fecf6f2073543d584481f0380ed8913e

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Citations in the corpus

Targeted Undersmoothing. (2017, methodology): “In addition, [25], [24] and [29] describe how bootstrapping can be used in conjunction with some of the previously cited techniques.”
Bootstrapping and sample splitting for high-dimensional, assumption-lean inference (2016, background): “, [14], [76], [36], [15], [26], [77], [72], [55].”
Generic Machine Learning Inference on Heterogenous Treatment Effects in Randomized Experiments (2017, unknown): “A related approach, which allows for simultaneous inference on many coefficients (for example, inference, on the coefficients corresponding to the interaction of the treatment with other variables) was first given in [14] using a Z-estimation framework, where the number of interactions can be very l”
A modern maximum-likelihood theory for high-dimensional logistic regression (2018, background): “In recent times, multiple approaches aimed to provide stronger simultaneous inference procedures for the ultra high-dimensional sparse linear model, see [36, 70, 73, 111].”
Breaking the curse of dimensionality in regression (2017, methodology): “However, exactly (or even approximately) sparse models are hardly appropriate for many modern scientific studies, see Hall et al. (2014); Ward (2009); Jin and Ke (2014); Pritchard (2001).”
High-dimensional econometrics and regularized GMM (2018, background): “[59] and [121] also consider bootstrap inference for many parameters estimated with debiased estimators in high-dimensional models building on [45].”
Beyond Gaussian approximation: Bootstrap for maxima of sums of independent random vectors (2017, methodology): “It can be also used in time series settings (Zhang and Wu, 2017a) and high-dimensional regression (Zhang and Zhang, 2014; Belloni et al., 2014, 2015; Zhang and Cheng, 2017; Dezeure et al., 2017).”