Comparison and anti-concentration bounds for maxima of Gaussian random vectors

By Victor Chernozhukov, D. Chetverikov, Kengo Kato. 2013

224 citations recorded by the source corpus.

Slepian and Sudakov–Fernique type inequalities, which compare expectations of maxima of Gaussian random vectors under certain restrictions on the covariance matrices, play an important role in probability theory, especially in empirical process and extreme value theories. Here we give explicit comparisons of expectations of smooth functions and distribution functions of maxima of Gaussian random vectors without any restriction on the covariance matrices. We also establish an anti-concentration inequality for the maximum of a Gaussian random vector, which derives a useful upper bound on the Lévy concentration function for the Gaussian maximum. The bound is dimension-free and applies to vectors with arbitrary covariance matrices. This anti-concentration inequality plays a crucial role in establishing bounds on the Kolmogorov distance between maxima of Gaussian random vectors. These results have immediate applications in mathematical statistics. As an example of application, we establish a conditional multiplier central limit theorem for maxima of sums of independent random vectors where the dimension of the vectors is possibly much larger than the sample size.

Subfields: Applied Mathematics, Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/602fe59fa33ec90185026fe4476bb5c18838e951

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Citations in the corpus

High-dimensional econometrics and regularized GMM (2018, methodology): “We therefore rely on the high-dimensional CLT developed in [45, 49, 50].”
Inference on Causal and Structural Parameters using Many Moment Inequalities (2013, methodology): “ther possible approach is to compare the distributions of max1≤j≤p √ nEn[Zij] and max1≤j≤p Yj directly, avoiding the comparison of distributions of the whole vectors √ nEn[Zi] and Y. Our recent work (Chernozhukov, Chetverikov, and Kato, 2013a, 2014b) shows that, under mild regularity conditions, the”
Bootstrapping and sample splitting for high-dimensional, assumption-lean inference (2016, methodology): “of Gaussian random vectors and Berry-Esseen bounds for polyhedral sets Now we collect some results that can be are derived from Chernozhukov et al. (2015), Chernozhukov et al. (2014) and Nazarov (2003).”
Orthogonal Machine Learning for Demand Estimation: High Dimensional Causal Inference in Dynamic Panels (2017, background): “As in [Chernozhukov et al., 2013a], the Gaussian approximation results above could be used not only for simultaneous confidence bands but also for multiple hypothesis testing using the step-down methods.”
Uniform post-selection inference for least absolute deviation regression and other Z-estimation problems (2013, background): “This inequality is less sharp than the dimension-free anticoncentration bound CaE[max1≤j≤p Yj ] proved in [18] for the case of max rectangles.”
Central limit theorems and bootstrap in high dimensions (2014, background): “See Lemma 8 in [17].”
Gaussian approximation of suprema of empirical processes (2012, background): “1 in [12] with the exception that we now apply Theorem 3, part (ii) instead of Theorem 3, part (i) from [11].”
Nonparametric modal regression (2014, methodology): “cases, the mixture-based method is often parametric (though can be semi-parametric or non-parametric in a sense, see Hunter and Young (2012); Huang et al. (2013) for mixture regression). In mixture-based methods, the EM algorithm is commonly applied o etstimate the parameters. Note that Chaganty and”
Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors (2012, methodology): “The following lemma, which is derived in [15], is a useful variation of standard maximal inequalities.”