Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors
We derive a Gaussian approximation result for the maximum of a sum of high-dimensional random vectors. Specifically, we establish conditions under which the distribution of the maximum is approximated by that of the maximum of a sum of the Gaussian random vectors with the same covariance matrices as...
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| Format: | Article |
| Language: | en_US |
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Institute of Mathematical Statistics
2014
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| Online Access: | http://hdl.handle.net/1721.1/85688 https://orcid.org/0000-0002-3250-6714 |
| _version_ | 1826198234424410112 |
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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 | We derive a Gaussian approximation result for the maximum of a sum of high-dimensional random vectors. Specifically, we establish conditions under which the distribution of the maximum is approximated by that of the maximum of a sum of the Gaussian random vectors with the same covariance matrices as the original vectors. This result applies when the dimension of random vectors (p) is large compared to the sample size (n); in fact, p can be much larger than n, without restricting correlations of the coordinates of these vectors. We also show that the distribution of the maximum of a sum of the random vectors with unknown covariance matrices can be consistently estimated by the distribution of the maximum of a sum of the conditional Gaussian random vectors obtained by multiplying the original vectors with i.i.d. Gaussian multipliers. This is the Gaussian multiplier (or wild) bootstrap procedure. Here too, p can be large or even much larger than n. These distributional approximations, either Gaussian or conditional Gaussian, yield a high-quality approximation to the distribution of the original maximum, often with approximation error decreasing polynomially in the sample size, and hence are of interest in many applications. We demonstrate how our Gaussian approximations and the multiplier bootstrap can be used for modern high-dimensional estimation, multiple hypothesis testing, and adaptive specification testing. All these results contain nonasymptotic bounds on approximation errors. |
| first_indexed | 2024-09-23T11:01:31Z |
| format | Article |
| id | mit-1721.1/85688 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T11:01:31Z |
| publishDate | 2014 |
| publisher | Institute of Mathematical Statistics |
| record_format | dspace |
| spelling | mit-1721.1/856882022-10-01T00:35:55Z Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors Chetverikov, Denis Kato, Kengo Chernozhukov, Victor V. Massachusetts Institute of Technology. Department of Economics Chernozhukov, Victor V. We derive a Gaussian approximation result for the maximum of a sum of high-dimensional random vectors. Specifically, we establish conditions under which the distribution of the maximum is approximated by that of the maximum of a sum of the Gaussian random vectors with the same covariance matrices as the original vectors. This result applies when the dimension of random vectors (p) is large compared to the sample size (n); in fact, p can be much larger than n, without restricting correlations of the coordinates of these vectors. We also show that the distribution of the maximum of a sum of the random vectors with unknown covariance matrices can be consistently estimated by the distribution of the maximum of a sum of the conditional Gaussian random vectors obtained by multiplying the original vectors with i.i.d. Gaussian multipliers. This is the Gaussian multiplier (or wild) bootstrap procedure. Here too, p can be large or even much larger than n. These distributional approximations, either Gaussian or conditional Gaussian, yield a high-quality approximation to the distribution of the original maximum, often with approximation error decreasing polynomially in the sample size, and hence are of interest in many applications. We demonstrate how our Gaussian approximations and the multiplier bootstrap can be used for modern high-dimensional estimation, multiple hypothesis testing, and adaptive specification testing. All these results contain nonasymptotic bounds on approximation errors. National Science Foundation (U.S.) 2014-03-17T19:58:22Z 2014-03-17T19:58:22Z 2013-12 2013-06 Article http://purl.org/eprint/type/JournalArticle 0090-5364 http://hdl.handle.net/1721.1/85688 Chernozhukov, Victor, Denis Chetverikov, and Kengo Kato. “Gaussian Approximations and Multiplier Bootstrap for Maxima of Sums of High-Dimensional Random Vectors.” Ann. Statist. 41, no. 6 (December 2013): 2786–2819. © Institute of Mathematical Statistics, 2013 https://orcid.org/0000-0002-3250-6714 en_US http://dx.doi.org/10.1214/13-AOS1161 The 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 Institute of Mathematical Statistics |
| spellingShingle | Chetverikov, Denis Kato, Kengo Chernozhukov, Victor V. Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors |
| title | Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors |
| title_full | Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors |
| title_fullStr | Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors |
| title_full_unstemmed | Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors |
| title_short | Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors |
| title_sort | gaussian approximations and multiplier bootstrap for maxima of sums of high dimensional random vectors |
| url | http://hdl.handle.net/1721.1/85688 https://orcid.org/0000-0002-3250-6714 |
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