Empirical and multiplier bootstraps for suprema of empirical processes of increasing complexity, and related Gaussian couplings
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Chernozhukov_Empirical.pdf
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Author(s)
Chetverikov, Denis
Kato, Kengo
Chernozhukov, Victor V
Date Issued
April 2016
Journal
Stochastic Processes and their Applications
Publisher
Elsevier BV
Citation
Chernozhukov, Victor, Denis Chetverikov, and Kengo Kato. “Empirical and Multiplier Bootstraps for Suprema of Empirical Processes of Increasing Complexity, and Related Gaussian Couplings.” Stochastic Processes and Their Applications 126, no. 12 (December 2016): 3632–3651.
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Author's final manuscript
Abstract
We derive strong approximations to the supremum of the non-centered empirical process indexed by a possibly unbounded VC-type class of functions by the suprema of the Gaussian and bootstrap processes. The bounds of these approximations are non-asymptotic, which allows us to work with classes of functions whose complexity increases with the sample size. The construction of couplings is not of the Hungarian type and is instead based on the Slepian–Stein methods and Gaussian comparison inequalities. The increasing complexity of classes of functions and non-centrality of the processes make the results useful for applications in modern nonparametric statistics (Giné and Nickl 2015), in particular allowing us to study the power properties of nonparametric tests using Gaussian and bootstrap approximations. Keywords: Coupling, Empirical process, Multiplier bootstrap process, Empirical bootstrap process, Gaussian approximation, Supremum
MIT Department
Massachusetts Institute of Technology. Department of Economics
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Creative Commons Attribution-NonCommercial-NoDerivs License
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DOI of Published Version
https://doi.org/10.1016/J.SPA.2016.04.009
