Victor Chernozhukov, Denis Chetverikov, Kengo Kato
arXiv 31 Dec 2012 · Mathematics — Statistics Theory · publishedThe Annals of Statistics (2013) · 341 citations (OpenAlex)
arXiv:1212.6906 · PDF · DOI · OpenAlex · Extracted main text
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.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Candès, E.J. and Tao, T (2007) The Dantzig selector: statistical estimation when $p$ is much larger than $n$ | 1.000 | 5 | 3 | 100% |
| 2 | Chernozhukov, V., Chetverikov, D., and Kato, K (2013) Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors self | 0.935 | 11 | 6 | 82% |
| 3 | Bickel, P., Ritov, Y. and Tsybakov, A (2009) Simultaneous analysis of Lasso and Dantzig selector | 0.874 | 7 | 2 | 100% |
| 4 | Romano, J., and Wolf, M (2005) Exact and approximate stepdown methods for multiple hypothesis testing | 0.874 | 5 | 2 | 100% |
| 5 | Talagrand, M (2003) Spin Glasses: A Challenge for Mathematicians | 0.843 | 4 | 3 | 75% |
| 6 | Chatterjee, S (2005) An error bound in the Sudakov-Fernique inequality | 0.737 | 3 | 3 | 67% |
| 7 | Chernozhukov, V., Chetverikov, D. and Kato, K (2012) Gaussian approximation of suprema of empirical processes self | 0.737 | 3 | 3 | 67% |
| 8 | Chernozhukov, V., Chetverikov, D. and Kato, K (2012) Comparison and anti-concentration bounds for maxima of Gaussian random vectors self | 0.721 | 8 | 6 | 38% |
| 9 | Dudley, R.M (1999) Uniform Central Limit Theorems | 0.644 | 3 | 2 | 67% |
| 10 | Arlot, S., Blanchard, G. and Roquain, E (2010) Some non-asymptotic results on resampling in high dimension II: multiple tests | 0.644 | 2 | 2 | 100% |
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