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Anti-concentration and honest, adaptive confidence bands

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
Modern construction of uniform confidence bands for non-parametric densities (and other functions) often relies on the classical Smirnov-Bickel-Rosenblatt (SBR) condition; see, for example, Giné and Nickl (2010). This condition requires the existence of a limit distribution of an extreme value type for the supremum of a studentized empirical process (equivalently, for the supremum of a Gaussian process with the same covariance function as that of the studentized empirical process). The principal contribution of this paper is to remove the need for this classical condition. We show that a considerably weaker sufficient condition is derived from an anti-concentration property of the supremum of the approximating Gaussian process, and we derive an inequality leading to such a property for separable Gaussian processes. We refer to the new condition as a generalized SBR condition. Our new result shows that the supremum does not concentrate too fast around any value. We then apply this result to derive a Gaussian multiplier bootstrap procedure for constructing honest confidence bands for nonparametric density estimators (this result can be applied in other nonparametetric problems as well). An essential advantage of our approach is that it applies generically even in those cases where the limit distribution of the supremum of the studentized empirical process does not exist (or is unknown). This is of particular importance in problems where resolution levels or other tuning parameters have been chosen in a data-driven fashion, which is needed for adaptive constructions of the confidence bands. Furthermore, our approach is asymptotically honest at a polynomial rate - namely, the error in coverage level converges to zero at a fast, polynomial speed (with respect to the sample size). In sharp contrast, the approach based on extreme value theory is asymptotically honest only at a logarithmic rate - the error converges to zero at a slow, logarithmic speed. Finally, of independent interest is our introduction of a new, practical version of Lepski's method, which computes the optimal, non-conservative resolution levels via a Gaussian multiplier bootstrap method.

Suggested Citation

  • Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2013. "Anti-concentration and honest, adaptive confidence bands," CeMMAP working papers CWP69/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    • Handle: RePEc:ifs:cemmap:69/13
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    References listed on IDEAS

    as
    1. Victor Chernozhukov & Sokbae Lee & Adam M. Rosen, 2013. "Intersection Bounds: Estimation and Inference," Econometrica, Econometric Society, vol. 81(2), pages 667-737, March.
    2. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2012. "Gaussian approximation of suprema of empirical processes," CeMMAP working papers CWP44/12, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    3. Newey, Whitney K., 1997. "Convergence rates and asymptotic normality for series estimators," Journal of Econometrics, Elsevier, vol. 79(1), pages 147-168, July.
    4. Y. Xia, 1998. "Bias‐corrected confidence bands in nonparametric regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(4), pages 797-811.
    5. Bissantz, Nicolai & Dümbgen, Lutz & Holzmann, Hajo & Munk, Axel, 2007. "Nonparametric confidence bands in deconvolution density estimation," Technical Reports 2007,03, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
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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. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2012. "Gaussian approximation of suprema of empirical processes," CeMMAP working papers CWP44/12, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    3. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2013. "Comparison and anti-concentration bounds for maxima of Gaussian random vectors," CeMMAP working papers CWP71/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    4. 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.
    5. 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 49/16, Institute for Fiscal Studies.

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