Anti-concentration and honest, adaptive confidence bands
Modern construction of uniform confidence bands for nonparametric densities (and other functions) often relies on the classical Smirnov–Bickel–Rosenblatt (SBR) condition; see, for example, Giné and Nickl [Probab. Theory Related Fields 143 (2009) 569–596]. This condition requires the existence of a l...
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Institute of Mathematical Statistics
2015
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| Online Access: | http://hdl.handle.net/1721.1/95958 https://orcid.org/0000-0002-3250-6714 |
| _version_ | 1811097887516393472 |
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| author | Chetverikov, Denis Kato, Kengo Chernozhukov, Victor V. |
| author2 | Massachusetts Institute of Technology. Department of Economics |
| author_facet | Massachusetts Institute of Technology. Department of Economics Chetverikov, Denis Kato, Kengo Chernozhukov, Victor V. |
| author_sort | Chetverikov, Denis |
| collection | MIT |
| description | Modern construction of uniform confidence bands for nonparametric densities (and other functions) often relies on the classical Smirnov–Bickel–Rosenblatt (SBR) condition; see, for example, Giné and Nickl [Probab. Theory Related Fields 143 (2009) 569–596]. 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 nonparametric 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. Finally, of independent interest is our introduction of a new, practical version of Lepski’s method, which computes the optimal, nonconservative resolution levels via a Gaussian multiplier bootstrap method. |
| first_indexed | 2024-09-23T17:06:32Z |
| format | Article |
| id | mit-1721.1/95958 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T17:06:32Z |
| publishDate | 2015 |
| publisher | Institute of Mathematical Statistics |
| record_format | dspace |
| spelling | mit-1721.1/959582022-10-03T10:26:49Z Anti-concentration and honest, adaptive confidence bands Chetverikov, Denis Kato, Kengo Chernozhukov, Victor V. Massachusetts Institute of Technology. Department of Economics Chernozhukov, Victor V. Modern construction of uniform confidence bands for nonparametric densities (and other functions) often relies on the classical Smirnov–Bickel–Rosenblatt (SBR) condition; see, for example, Giné and Nickl [Probab. Theory Related Fields 143 (2009) 569–596]. 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 nonparametric 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. Finally, of independent interest is our introduction of a new, practical version of Lepski’s method, which computes the optimal, nonconservative resolution levels via a Gaussian multiplier bootstrap method. National Science Foundation (U.S.) 2015-03-11T19:02:29Z 2015-03-11T19:02:29Z 2014-09 2014-04 Article http://purl.org/eprint/type/JournalArticle 0090-5364 http://hdl.handle.net/1721.1/95958 Chernozhukov, Victor, Denis Chetverikov, and Kengo Kato. “Anti-Concentration and Honest, Adaptive Confidence Bands.” Ann. Statist. 42, no. 5 (October 2014): 1787–1818. © 2014 Institute of Mathematical Statistics https://orcid.org/0000-0002-3250-6714 en_US http://dx.doi.org/10.1214/14-aos1235 Annals of Statistics Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Institute of Mathematical Statistics arXiv |
| spellingShingle | Chetverikov, Denis Kato, Kengo Chernozhukov, Victor V. Anti-concentration and honest, adaptive confidence bands |
| title | Anti-concentration and honest, adaptive confidence bands |
| title_full | Anti-concentration and honest, adaptive confidence bands |
| title_fullStr | Anti-concentration and honest, adaptive confidence bands |
| title_full_unstemmed | Anti-concentration and honest, adaptive confidence bands |
| title_short | Anti-concentration and honest, adaptive confidence bands |
| title_sort | anti concentration and honest adaptive confidence bands |
| url | http://hdl.handle.net/1721.1/95958 https://orcid.org/0000-0002-3250-6714 |
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