Simultaneous Inference for High-Dimensional Linear Models

By Xianyang Zhang, Guang Cheng. 2016

150 citations recorded by the source corpus.

ABSTRACT This article proposes a bootstrap-assisted procedure to conduct simultaneous inference for high-dimensional sparse linear models based on the recent desparsifying Lasso estimator. Our procedure allows the dimension of the parameter vector of interest to be exponentially larger than sample size, and it automatically accounts for the dependence within the desparsifying Lasso estimator. Moreover, our simultaneous testing method can be naturally coupled with the margin screening to enhance its power in sparse testing with a reduced computational cost, or with the step-down method to provide a strong control for the family-wise error rate. In theory, we prove that our simultaneous testing procedure asymptotically achieves the prespecified significance level, and enjoys certain optimality in terms of its power even when the model errors are non-Gaussian. Our general theory is also useful in studying the support recovery problem. To broaden the applicability, we further extend our main results to generalized linear models with convex loss functions. The effectiveness of our methods is demonstrated via simulation studies. Supplementary materials for this article are available online.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ec30899ea0e5a2edf501e7319c285c15d1100022

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Citations in the corpus

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].”
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].”
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).”
Breaking the curse of dimensionality in regression (2017, methodology): “We consider the test proposed in Zhang and Cheng (2016) based on the de-sparsified Lasso.”
Bootstrapping and sample splitting for high-dimensional, assumption-lean inference (2016, background): “, [14], [76], [36], [15], [26], [77], [72], [55].”