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Improved Central Limit Theorem and bootstrap approximations in high dimensions

Author

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  • Victor Chernozhukov
  • Denis Chetverikov
  • Kengo Kato
  • Yuta Koike

Abstract

This paper deals with the Gaussian and bootstrap approximations to the distribution of the max statistic in high dimensions. This statistic takes the form of the maximum over components of the sum of independent random vectors and its distribution plays a key role in many high-dimensional econometric problems. Using a novel iterative randomized Lindeberg method, the paper derives new bounds for the distributional approximation errors. These new bounds substantially improve upon existing ones and simultaneously allow for a larger class of bootstrap methods.

Suggested Citation

  • Victor Chernozhukov & Denis Chetverikov & Kengo Kato & Yuta Koike, 2019. "Improved Central Limit Theorem and bootstrap approximations in high dimensions," Papers 1912.10529, arXiv.org, revised May 2022.
    • Handle: RePEc:arx:papers:1912.10529
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    File URL: http://arxiv.org/pdf/1912.10529
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    References listed on IDEAS

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    1. Andrew Chesher & Adam Rosen, 2018. "Generalized instrumental variable models, methods, and applications," CeMMAP working papers CWP43/18, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
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    3. Hansen, Peter Reinhard, 2005. "A Test for Superior Predictive Ability," Journal of Business & Economic Statistics, American Statistical Association, vol. 23, pages 365-380, October.
    4. Victor Chernozhukov & Denis Chetverikov & Kengo Kato & Aureo de Paula, 2019. "Inference on Causal and Structural Parameters using Many Moment Inequalities," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 86(5), pages 1867-1900.
    5. Alexandre Belloni & Victor Chernozhukov & Denis Chetverikov & Christian Hansen & Kengo Kato, 2018. "High-dimensional econometrics and regularized GMM," CeMMAP working papers CWP35/18, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    6. Joseph P. Romano & Azeem M. Shaikh & Michael Wolf, 2014. "A Practical Two‐Step Method for Testing Moment Inequalities," Econometrica, Econometric Society, vol. 82, pages 1979-2002, September.
    7. Chetverikov, Denis, 2018. "Adaptive Tests Of Conditional Moment Inequalities," Econometric Theory, Cambridge University Press, vol. 34(1), pages 186-227, February.
    8. Chetverikov, Denis & Wilhelm, Daniel & Kim, Dongwoo, 2021. "An Adaptive Test Of Stochastic Monotonicity," Econometric Theory, Cambridge University Press, vol. 37(3), pages 495-536, June.
    9. 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.
    10. Peter R. Hansen & Asger Lunde & James M. Nason, 2011. "The Model Confidence Set," Econometrica, Econometric Society, vol. 79(2), pages 453-497, March.
    11. Andrews, Donald W.K. & Shi, Xiaoxia, 2014. "Nonparametric inference based on conditional moment inequalities," Journal of Econometrics, Elsevier, vol. 179(1), pages 31-45.
    12. Chetverikov, Denis, 2019. "Testing Regression Monotonicity In Econometric Models," Econometric Theory, Cambridge University Press, vol. 35(4), pages 729-776, August.
    13. Nick Koning & Paul Bekker, 2019. "Exact Testing of Many Moment Inequalities Against Multiple Violations," Papers 1904.12775, arXiv.org, revised Jun 2020.
    14. Halbert White, 2000. "A Reality Check for Data Snooping," Econometrica, Econometric Society, vol. 68(5), pages 1097-1126, September.
    15. repec:cwl:cwldpp:1840rr is not listed on IDEAS
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    Cited by:

    1. Magne Mogstad & Joseph P. Romano & Azeem Shaikh & Daniel Wilhelm, 2020. "Inference for Ranks with Applications to Mobility across Neighborhoods and Academic Achievement across Countries," NBER Working Papers 26883, National Bureau of Economic Research, Inc.
    2. Bakalli, Gaetan & Guerrier, Stéphane & Scaillet, Olivier, 2023. "A penalized two-pass regression to predict stock returns with time-varying risk premia," Journal of Econometrics, Elsevier, vol. 237(2).
    3. Anders Bredahl Kock & David Preinerstorfer, 2023. "A remark on moment-dependent phase transitions in high-dimensional Gaussian approximations," Papers 2310.12863, arXiv.org, revised Feb 2024.
    4. Cheng, Guanghui & Liu, Zhi & Peng, Liuhua, 2022. "Gaussian approximations for high-dimensional non-degenerate U-statistics via exchangeable pairs," Statistics & Probability Letters, Elsevier, vol. 182(C).
    5. Matias D. Cattaneo & Ricardo P. Masini & William G. Underwood, 2022. "Yurinskii's Coupling for Martingales," Papers 2210.00362, arXiv.org, revised Mar 2024.
    6. Matias D. Cattaneo & Rajita Chandak & Michael Jansson & Xinwei Ma, 2022. "Boundary Adaptive Local Polynomial Conditional Density Estimators," Papers 2204.10359, arXiv.org, revised Dec 2023.
    7. Chang, Jinyuan & Jiang, Qing & Shao, Xiaofeng, 2023. "Testing the martingale difference hypothesis in high dimension," Journal of Econometrics, Elsevier, vol. 235(2), pages 972-1000.
    8. 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.
    9. Victor Chernozhukov & Denis Chetverikov & Kengo Kato & Yuta Koike, 2022. "High-dimensional Data Bootstrap," Papers 2205.09691, arXiv.org.
    10. Kojevnikov, Denis & Song, Kyungchul, 2022. "A Berry–Esseen bound for vector-valued martingales," Statistics & Probability Letters, Elsevier, vol. 186(C).

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