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Central limit theorems and multiplier bootstrap when p is much larger than n

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

We derive a central limit theorem for the maximum of a sum of high dimensional random vectors. More precisely, we establish conditions under which the distribution of the maximum is approximated by the maximum of a sum of the Gaussian random vectors with the same covariance matrices as the original vectors. The key innovation of our result is that it applies even if the dimension of random vectors (p) is much larger than the sample size (n). In fact, the growth of p could be exponential in some fractional power of n. We also show that the distribution of the maximum of a sum of the Gaussian random vectors with unknown covariance matrices can be estimated by the distribution of the maximum of the (conditional) Gaussian process obtained by multiplying the original vectors with i.i.d. Gaussian multipliers. We call this procedure the “multiplier bootstrap”. Here too, the growth of p could be exponential in some fractional power of n. We prove that our distributional approximations, either Gaussian or conditional Gaussian, yield a high-quality approximation for the distribution of the original maximum, often with at most a polynomial approximation error. These results are of interest in numerous econometric and statistical applications. In particular, we demonstrate how our central limit theorem and the multiplier bootstrap can be used for high dimensional estimation, multiple hypothesis testing, and adaptive specification testing. All of our results contain non-asymptotic bounds on approximation errors.

Suggested Citation

  • Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2012. "Central limit theorems and multiplier bootstrap when p is much larger than n," CeMMAP working papers CWP45/12, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  • Handle: RePEc:ifs:cemmap:45/12
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    File URL: http://www.cemmap.ac.uk/wps/cwp451212.pdf
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    References listed on IDEAS

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    1. 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.
    2. Joseph P. Romano & Michael Wolf, 2005. "Exact and Approximate Stepdown Methods for Multiple Hypothesis Testing," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 94-108, March.
    3. Eric Gautier & Alexandre Tsybakov, 2011. "High-Dimensional Instrumental Variables Regression and Confidence Sets," Working Papers 2011-13, Center for Research in Economics and Statistics.
    4. Horowitz, Joel L & Spokoiny, Vladimir G, 2001. "An Adaptive, Rate-Optimal Test of a Parametric Mean-Regression Model against a Nonparametric Alternative," Econometrica, Econometric Society, vol. 69(3), pages 599-631, May.
    5. Fan, Jianqing & Hall, Peter & Yao, Qiwei, 2007. "To How Many Simultaneous Hypothesis Tests Can Normal, Student's t or Bootstrap Calibration Be Applied?," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 1282-1288, December.
    6. A. Belloni & V. Chernozhukov & L. Wang, 2011. "Square-root lasso: pivotal recovery of sparse signals via conic programming," Biometrika, Biometrika Trust, vol. 98(4), pages 791-806.
    7. Alquier, Pierre & Hebiri, Mohamed, 2011. "Generalization of ℓ1 constraints for high dimensional regression problems," Statistics & Probability Letters, Elsevier, vol. 81(12), pages 1760-1765.
    8. Emmanuel Guerre & Pascal Lavergne, 2004. "Data-Driven Rate-Optimal Specification Testing In Regression Models," Econometrics 0411008, EconWPA.
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