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Central limit theorems and bootstrap in high dimensions

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  • Victor Chernozhukov
  • Denis Chetverikov
  • Kengo Kato

Abstract

In this paper, we derive central limit and bootstrap theorems for probabilities that centered high-dimensional vector sums hit rectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for the probabilities that a root-n rescaled sample average of Xi is in A, where X1,..., Xn are independent random vectors in Rp and A is a rectangle, or, more generally, a sparsely convex set, and show that the approximation error converges to zero even if p=pn-> infinity and p>>n; in particular, p can be as large as O(e^(Cn^c)) for some constants c,C>0. The result holds uniformly over all rectangles, or more generally, sparsely convex sets, and does not require any restrictions on the correlation among components of Xi. Sparsely convex sets are sets that can be represented as intersections of many convex sets whose indicator functions depend nontrivially only on a small subset of their arguments, with rectangles being a special case.

Suggested Citation

  • Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2014. "Central limit theorems and bootstrap in high dimensions," CeMMAP working papers 49/14, Institute for Fiscal Studies.
  • Handle: RePEc:azt:cemmap:49/14
    DOI: 10.1920/wp.cem.2014.4914
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    References listed on IDEAS

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    1. Radosław Adamczak, 2010. "A Few Remarks on the Operator Norm of Random Toeplitz Matrices," Journal of Theoretical Probability, Springer, vol. 23(1), pages 85-108, March.
    2. Yosef Rinott & Vladimir Rotar, 2000. "Normal approximations by Stein's method," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 23(1), pages 15-29.
    3. 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 & Denis Chetverikov & Kengo Kato, 2013. "Testing Many Moment Inequalities," CeMMAP working papers 65/13, Institute for Fiscal Studies.
    2. Chernozhukov, Victor & Chetverikov, Denis & Kato, Kengo, 2016. "Empirical and multiplier bootstraps for suprema of empirical processes of increasing complexity, and related Gaussian couplings," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3632-3651.
    3. 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.
    4. Victor Chernozhukov & Denis Chetverikov & Mert Demirer & Esther Duflo & Christian Hansen & Whitney K. Newey, 2016. "Double machine learning for treatment and causal parameters," CeMMAP working papers CWP49/16, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    5. Fabian Dunker & Konstantin Eckle & Katharina Proksch & Johannes Schmidt-Hieber, 2017. "Tests for qualitative features in the random coefficients model," Papers 1704.01066, arXiv.org, revised Mar 2018.
    6. Ruben Dezeure & Peter Bühlmann & Cun-Hui Zhang, 2017. "High-dimensional simultaneous inference with the bootstrap," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(4), pages 685-719, December.
    7. Denis Chetverikov & . ., 2016. "On cross-validated Lasso," CeMMAP working papers CWP47/16, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    8. Denis Chetverikov & . ., 2016. "On cross-validated Lasso," CeMMAP working papers 47/16, Institute for Fiscal Studies.
    9. Ebert, Johannes & Spokoiny, Vladimir & Suvorikova, Alexandra, 2018. "Construction of Non-asymptotic Confidence Sets in 2 -Wasserstein Space," IRTG 1792 Discussion Papers 2018-025, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    10. Naumov, A. & Spokoiny, V. & Ulyanovk, V., 2018. "Bootstrap Confidence Sets for Spectral Projectors of Sample Covariance," IRTG 1792 Discussion Papers 2018-024, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".

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