Central limit theorems and multiplier bootstrap when p is much larger than n
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.
|Date of creation:||18 Dec 2012|
|Contact details of provider:|| Postal: The Institute for Fiscal Studies 7 Ridgmount Street LONDON WC1E 7AE|
Phone: (+44) 020 7291 4800
Fax: (+44) 020 7323 4780
Web page: http://cemmap.ifs.org.uk
More information through EDIRC
|Order Information:|| Postal: The Institute for Fiscal Studies 7 Ridgmount Street LONDON WC1E 7AE|
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- 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.
- Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2013. "Gaussian approximation of suprema of empirical processes," CeMMAP working papers CWP75/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
- Emmanuel Guerre & Pascal Lavergne, 2004. "Data-Driven Rate-Optimal Specification Testing In Regression Models," Econometrics 0411008, EconWPA.
- 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.
- Joseph Romano & Michael Wolf, 2003.
"Exact and approximate stepdown methods for multiple hypothesis testing,"
Economics Working Papers
727, Department of Economics and Business, Universitat Pompeu Fabra.
- 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.
- 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.
- Alquier, Pierre & Hebiri, Mohamed, 2011. "Generalization of ℓ1 constraints for high dimensional regression problems," Statistics & Probability Letters, Elsevier, vol. 81(12), pages 1760-1765.
- Eric Gautier & Alexandre Tsybakov, 2011.
"High-Dimensional Instrumental Variables Regression and Confidence Sets,"
2011-13, Centre de Recherche en Economie et Statistique.
- Eric Gautier & Alexandre Tsybakov, 2014. "High-dimensional instrumental variables regression and confidence sets," Working Papers hal-00591732, HAL.
- 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.
When requesting a correction, please mention this item's handle: RePEc:ifs:cemmap:45/12. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Emma Hyman)
If references are entirely missing, you can add them using this form.