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Empirical and multiplier bootstraps for suprema of empirical processes of increasing complexity, and related Gaussian couplings

Author

Listed:
  • Victor Chernozhukov

    (Institute for Fiscal Studies and MIT)

  • Denis Chetverikov

    (Institute for Fiscal Studies and UCLA)

  • Kengo Kato

    (Institute for Fiscal Studies)

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 [14]), in particular allowing us to study the power properties of nonparametric tests using Gaussian and bootstrap approximations.

Suggested Citation

  • Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2016. "Empirical and multiplier bootstraps for suprema of empirical processes of increasing complexity, and related Gaussian couplings," CeMMAP working papers CWP38/16, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  • Handle: RePEc:ifs:cemmap:38/16
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    References listed on IDEAS

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    1. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2014. "Central limit theorems and bootstrap in high dimensions," CeMMAP working papers CWP49/14, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    2. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2012. "Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors," Papers 1212.6906, arXiv.org, revised Jan 2018.
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    Cited by:

    1. Kato, Kengo & Sasaki, Yuya, 2019. "Uniform confidence bands for nonparametric errors-in-variables regression," Journal of Econometrics, Elsevier, vol. 213(2), pages 516-555.
    2. Jun Ma & Vadim Marmer & Artyom Shneyerov & Pai Xu, 2019. "Monotonicity-Constrained Nonparametric Estimation and Inference for First-Price Auctions," Papers 1909.12974, arXiv.org.
    3. Victor Chernozhukov & Ivan Fernandez-Val & Martin Weidner, 2018. "Network and panel quantile effects via distribution regression," CeMMAP working papers CWP21/18, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    4. Ma, Jun & Marmer, Vadim & Shneyerov, Artyom, 2019. "Inference for first-price auctions with Guerre, Perrigne, and Vuong’s estimator," Journal of Econometrics, Elsevier, vol. 211(2), pages 507-538.
    5. Kato, Kengo & Kurisu, Daisuke, 2020. "Bootstrap confidence bands for spectral estimation of Lévy densities under high-frequency observations," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1159-1205.
    6. Kato, Kengo & Sasaki, Yuya, 2018. "Uniform confidence bands in deconvolution with unknown error distribution," Journal of Econometrics, Elsevier, vol. 207(1), pages 129-161.

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    More about this item

    Keywords

    coupling; empirical process; multiplier bootstrap process; empirical bootstrap process; Gaussian approximation; supremum;
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