Central limit theorems and bootstrap in high dimensions

By Victor Chernozhukov, D. Chetverikov, Kengo Kato. 2014

342 citations recorded by the source corpus.

In this paper, we derive central limit and bootstrap theorems for probabilities that centered high-dimensional vector sums hit rectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for the probabilities that a root-n rescaled sample average of Xi is in A, where X1,..., Xn are independent random vectors in Rp and A is a rectangle, or, more generally, a sparsely convex set, and show that the approximation error converges to zero even if p=pn-> infinity and p>>n; in particular, p can be as large as O(e^(Cn^c)) for some constants c,C>0. The result holds uniformly over all rectangles, or more generally, sparsely convex sets, and does not require any restrictions on the correlation among components of Xi. Sparsely convex sets are sets that can be represented as intersections of many convex sets whose indicator functions depend nontrivially only on a small subset of their arguments, with rectangles being a special case.

Subfields: Artificial Intelligence, Statistics and Probability

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Citations in the corpus

Double/De-Biased Machine Learning of Global and Local Parameters Using Regularized Riesz Representers (2018, methodology): “, 2014d] or high-dimensional central limit theorems [Chernozhukov et al., 2017], under mild moment conditions, without requiring sub-Gaussianity.”
Estimation and Inference about Heterogeneous Treatment Effects in High-Dimensional Dynamic Panels. (2017, methodology): “ler model than ours. In addition, we contribute to a growing literature on high-dimensional Gaussian approximation (Chernozhukov et al. (2013), Chernozhukov et al. (2014), Chernozhukov et al. (2015), Chernozhukov et al. (2017b), Zhang and Wu (2017), Chernozhukov et al. (2019a), Chernozhukov et al. (”
Beyond Gaussian approximation: Bootstrap for maxima of sums of independent random vectors (2017, methodology): “This issue is again resolved through Lemma 2.”
Finite-sample Analysis of M-estimators using Self-concordance (2018, methodology): “The main highlight of [14] is the ability to handle the regime of exponentially large dimensionality, with respect to the sample size, due to the special structure of the statistics under study.”
Change-point detection in high-dimensional covariance structure (2016, methodology): “By employing the Gaussian comparison result provided by [10] and [9] obtain similarity of joint distributions of the norms of ζ and η, which in combination with the steps 1 and 2 yields similarity of joint distributions of An and A ♭ n.”
Breaking the curse of dimensionality in regression (2017, methodology): “This, together with (67), (68), (69) and (70), allows us to apply Proposition 2.1 of Chernozhukov et al. (2014) to the conditional probability measure P(· | Fn).”
Valid post-selection inference in model-free linear regression (2020, methodology): “In this subsection, we consider the consistency of the multiplier bootstrap based on Section 4.1 of Chernozhukov et al. (2017).”
High-dimensional simultaneous inference with the bootstrap (2016, methodology): “(Extensions to the classical bootstrap might be rather straightforward using the results from Chernozhukov et al. 2014.)”
Gaussian and bootstrap approximations for high-dimensional U-statistics and their applications (2016, methodology): “The Gaussian and bootstrap approximations for the maxima of sums of high-dimensional independent random vectors were considered in [24, 22].”
Improved central limit theorem and bootstrap approximations in high dimensions (2019, neutral): “te approximations to the distribution of Tn, allowing for the high-dimensional p≫ ncase. Toward this goal, the ﬁrst three authors of this paper obtained the following Gaussian approximation result in [11, 14]. Let G= (G 1,...,Gp)′ be a Gaussian random vector in Rpwith mean µ= (µ 1,...,µp)′ and varia”
Detailed proof of Nazarov's inequality (2017, methodology): “Unless otherwise stated, we shall follow the notational convention used in [1].”
Testing Many Moment Inequalities (2013, background): “Thus, both our paper and Zhang and Cheng (2014) extend Gaussian approximation and bootstrap results of Chernozhukov, Chetverikov, and Kato (2013a) to the case of dependent data but under different dependence conditions (that do not nest each other). The results obtained in these two papers are stron”