On the dependence of the Berry–Esseen bound on dimension

By V. Bentkus. 2003

198 citations recorded by the source corpus.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/82037da94c081ef3b6d41f8bf4df3438dc9861f7

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

Testing Many Moment Inequalities (2013, methodology): “One possible approach is to use a Berry-Esseen theorem that provides a suitable non-asymptotic bound between the distributions of √ nEn[Zi] and Y ; see, for example, Götze (1991) and Bentkus (2003). However, such Berry-Esseen bounds require p to be small in comparison with n in order to guarantee th”
High-dimensional random geometric graphs and their clique number (2011, methodology): “In particular, we use the following multivariate Berry-Esséen inequality (see Bentkus [4], Raič [17]) for convex sets.”
Chi-square approximation by Stein's method with application to Pearson's statistic (2015, background): “1 of [3] in place of Theorem 1.”
MULTIVARIATE NORMAL APPROXIMATION USING EXCHANGEABLE PAIRS (2007, unknown): “Bentkus (2003), where one can also find extensive pointers to the literature.”
Bolasso: model consistent Lasso estimation through the bootstrap (2008, background): “We also consider multivariate Berry-Esseen inequalities (Bentkus, 2003); the probability P(nq ∈ C) can be estimated as P(t ∈ C) where t is normal with mean zero and covariance matrix σ(2)Q.”
A multivariate Berry–Esseen theorem with explicit constants (2018, methodology): “However, this is still of larger order than Bentkus’s result.”
Central limit theorems and bootstrap in high dimensions (2014, background): “In particular, Bentkus [7] established one of the sharpest results in this direction which states that when X 1 , . . . , X n are i.i.d. with E[ X i X ′ i ] = I ( I denotes the p × p identity matrix), where C p ( A ) is a constant that depends only on p and A ; for example, C p ( A ) is bounded by a”
Large ball probabilities, Gaussian comparison and anti-concentration (2017, background): “Applications of smoothing inequalities require to evaluate the probability of hitting the vicinity of a convex set, see e.g. Bentkus (2003), Bentkus (2005).”
Gaussian and bootstrap approximations for high-dimensional U-statistics and their applications (2016, background): “Similar observations have been made in the existing literature [61, 5] for the normalized sample mean vectors of iid mean-zero random vectors Xi ∈ Rd, which corresponds to a U-statistic with the linear kernel h(x1, x2) = (x1 + x2)/2.”
Inference on Causal and Structural Parameters using Many Moment Inequalities (2013, methodology): “ue for the test. One possible approach is to use a Berry-Esseen theorem that provides a suitable non-asymptotic bound between the distributions of √ nEn[Zi] and Y; see, for example, Go¨tze (1991) and Bentkus (2003). However, such Berry-Esseen bounds require pto be small in comparison with nin order ”
Minimax adaptive tests for the Functional Linear model (2012, methodology): “ j-th component is (j) i i. We note kW ik k its Euclidean norm. By Assumption B:1, we have E kW ik3 k k3=2E 4 3=4 sup 1jk E h ((j))4 i 3=4 : Applying the second part of Theorem 1.1 in Bentkus [4], we obtain sup x>0 p P E k nA k nk 2 x ˜ k(x=˙2) 4 C k 3=2 p n 3=4 ˙3 sup 1 j k E h ((j))4 i 4 : We”
High-dimensional econometrics and regularized GMM (2018, background): “For example, [28, 29] show that a result like (2.”
Uniform post-selection inference for least absolute deviation regression and other Z-estimation problems (2013, background): “(i) The bound (6) should be contrasted with Bentkus’s [7] bound (2).”
A high-dimensional CLT in W2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {W}_2$$\end{document} distance wi (2017, background): “The best convergence rate seems to be due to Bentkus [2].”
Nearly optimal central limit theorem and bootstrap approximations in high dimensions (2020, methodology): “hen Xi’s have independent components with vanishing odd moments. Our results are built using the exchangeable pair approach coupled with the SlepianStein method and employ ideas of many authors, e.g. [7, 28, 3, 8, 11, 36, 12, 15, 23, 32, 31], but the key technical tool behind our results is a set of”
Comparison and anti-concentration bounds for maxima of Gaussian random vectors (2013, background): “4) noted in [3], which is based on [2], and from the fact that the sets of the form Amax(t) = {x ∈ R p : max1≤j≤p xj ≤ t} are convex.”
A Lyapunov-type Bound in Rd (2005, methodology): “The following smoothing lemma is contained in [3].”
Distributed Statistical Estimation and Rates of Convergence in Normal Approximation (2017, methodology): “The following multivariate version of the Berry-Esseen Theorem for convex sets has been established by Bentkus (2003).”
Beyond Gaussian approximation: Bootstrap for maxima of sums of independent random vectors (2017, background): “In closely related settings, Giné and Zinn (1990) proved the consistency of bootstrap for Donsker classes of functions, Nagaev (1976), Senatov (1980), Sazonov (1981), Gotze (1991) and Bentkus (1986, 2003) for convex sets when n ≥ p7/2, and Zhilova (2016) for Euclidean balls.”