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

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  • Victor Chernozhukov
  • Denis Chetverikov
  • Kengo Kato

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

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  • 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 38/16, Institute for Fiscal Studies.
    • Handle: RePEc:azt:cemmap:38/16
    DOI: 10.1920/wp.cem.2016.3816
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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. 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.
    2. Alexandre Belloni & Victor Chernozhukov & Abhishek Kaul, 2017. "Confidence bands for coefficients in high dimensional linear models with error-in-variables," CeMMAP working papers CWP22/17, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    3. Belloni, Alexandre & Chernozhukov, Victor & Chetverikov, Denis & Fernández-Val, Iván, 2019. "Conditional quantile processes based on series or many regressors," Journal of Econometrics, Elsevier, vol. 213(1), pages 4-29.
    4. Matias D. Cattaneo & Richard K. Crump & Weining Wang, 2022. "Beta-Sorted Portfolios," Papers 2208.10974, arXiv.org, revised Jul 2023.
    5. Babii, Andrii, 2020. "Honest Confidence Sets In Nonparametric Iv Regression And Other Ill-Posed Models," Econometric Theory, Cambridge University Press, vol. 36(4), pages 658-706, August.
    6. Kurisu, Daisuke & Otsu, Taisuke, 2022. "On linearization of nonparametric deconvolution estimators for repeated measurements model," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    7. 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.
    8. Jun Ma & Vadim Marmer & Artyom Shneyerov & Pai Xu, 2021. "Monotonicity-constrained nonparametric estimation and inference for first-price auctions," Econometric Reviews, Taylor & Francis Journals, vol. 40(10), pages 944-982, November.
    9. Alexandre Belloni & Mingli Chen & Victor Chernozhukov, 2016. "Quantile Graphical Models: Prediction and Conditional Independence with Applications to Systemic Risk," Papers 1607.00286, arXiv.org, revised Oct 2019.
    10. Ma, Jun & Marmer, Vadim & Yu, Zhengfei, 2023. "Inference on individual treatment effects in nonseparable triangular models," Journal of Econometrics, Elsevier, vol. 235(2), pages 2096-2124.
    11. Yuta Koike, 2023. "High-Dimensional Central Limit Theorems for Homogeneous Sums," Journal of Theoretical Probability, Springer, vol. 36(1), pages 1-45, March.
    12. Kato, Kengo & Sasaki, Yuya, 2019. "Uniform confidence bands for nonparametric errors-in-variables regression," Journal of Econometrics, Elsevier, vol. 213(2), pages 516-555.
    13. Dong, Hao & Taylor, Luke, 2022. "Nonparametric Significance Testing In Measurement Error Models," Econometric Theory, Cambridge University Press, vol. 38(3), pages 454-496, June.
    14. 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.
    15. Shi, Chengchun & Luo, Shikai & Zhu, Hongtu & Song, Rui, 2021. "An online sequential test for qualitative treatment effects," LSE Research Online Documents on Economics 112521, London School of Economics and Political Science, LSE Library.
    16. Peccati, Giovanni & Turchi, Nicola, 2023. "The discrepancy between min–max statistics of Gaussian and Gaussian-subordinated matrices," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 315-341.
    17. Kato, Kengo & Sasaki, Yuya, 2018. "Uniform confidence bands in deconvolution with unknown error distribution," Journal of Econometrics, Elsevier, vol. 207(1), pages 129-161.
    18. Daisuke Kurisu & Taisuke Otsu, 2021. "On linearization of nonparametric deconvolution estimators for repeated measurements model," STICERD - Econometrics Paper Series 615, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    19. Belloni, Alexandre & Hansen, Christian & Newey, Whitney, 2022. "High-dimensional linear models with many endogenous variables," Journal of Econometrics, Elsevier, vol. 228(1), pages 4-26.
    20. Byunghoon Kang, 2019. "Inference in Nonparametric Series Estimation with Specification Searches for the Number of Series Terms," Papers 1909.12162, arXiv.org, revised Feb 2020.

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