Comparison and anti-concentration bounds for maxima of Gaussian random vectors
TLDR
In this paper, the authors give explicit comparisons of expectations of smooth functions and distribution functions of maxima of Gaussian random vectors without any restriction on the covariance matrices, and establish an anti-concentration inequality for the maximum of a Gaussian vector, which derives a useful upper bound on the Levy concentration function for the Gaussian maximum.Abstract:
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 Levy 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.read more
Citations
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ReportDOI
Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors
TL;DR: It is demonstrated how the Gaussian approximations and the multiplier bootstrap can be used for modern high dimensional estimation, multiple hypothesis testing, and adaptive specification testing.
ReportDOI
Gaussian approximation of suprema of empirical processes
TL;DR: An abstract approximation theorem that is applicable to a wide variety of problems, primarily in statistics, is proved and the bound in the main approximation theorem is non-asymptotic and the theorem does not require uniform boundedness of the class of functions.
Supplement to \gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors"
TL;DR: The Gaussian approximations and the multiplier bootstrap can be used for modern high-dimensional estimation, multiple hypothesis testing, and adaptive specification testing and contain nonasymptotic bounds on approximation errors.
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Uniform post-selection inference for least absolute deviation regression and other Z-estimation problems
TL;DR: In this paper, Neyman's orthogonal score test is applied to a high-dimensional sparse median regression model with homoscedastic errors and uniformly valid confidence regions for regression coefficients are developed.
References
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Book
Weak Convergence and Empirical Processes: With Applications to Statistics
TL;DR: In this article, the authors define the Ball Sigma-Field and Measurability of Suprema and show that it is possible to achieve convergence almost surely and in probability.
BookDOI
Weak Convergence and Empirical Processes
TL;DR: This chapter discusses Convergence: Weak, Almost Uniform, and in Probability, which focuses on the part of Convergence of the Donsker Property which is concerned with Uniformity and Metrization.
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The Dantzig selector: Statistical estimation when p is much larger than n
Emmanuel J. Candès,Terence Tao +1 more
TL;DR: In many important statistical applications, the number of variables or parameters p is much larger than the total number of observations n as discussed by the authors, and it is possible to estimate β reliably based on the noisy data y.
Journal ArticleDOI
Estimation of the Mean of a Multivariate Normal Distribution
TL;DR: In this article, an unbiased estimate of risk is obtained for an arbitrary estimate, and certain special classes of estimates are then discussed, such as smoothing by using moving averages and trimmed analogs of the James-Stein estimate.
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Statistics for High-Dimensional Data: Methods, Theory and Applications
Peter Bhlmann,Sara van de Geer +1 more
TL;DR: This book presents a detailed account of recently developed approaches, including the Lasso and versions of it for various models, boosting methods, undirected graphical modeling, and procedures controlling false positive selections.