The Annals of Statistics

Some nonasymptotic results on resampling in high dimension, I: Confidence regions

Sylvain Arlot, Gilles Blanchard, and Etienne Roquain

Full-text: Open access


We study generalized bootstrap confidence regions for the mean of a random vector whose coordinates have an unknown dependency structure. The random vector is supposed to be either Gaussian or to have a symmetric and bounded distribution. The dimensionality of the vector can possibly be much larger than the number of observations and we focus on a nonasymptotic control of the confidence level, following ideas inspired by recent results in learning theory. We consider two approaches, the first based on a concentration principle (valid for a large class of resampling weights) and the second on a resampled quantile, specifically using Rademacher weights. Several intermediate results established in the approach based on concentration principles are of interest in their own right. We also discuss the question of accuracy when using Monte Carlo approximations of the resampled quantities.

Article information

Ann. Statist., Volume 38, Number 1 (2010), 51-82.

First available in Project Euclid: 31 December 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G15: Tolerance and confidence regions
Secondary: 62G09: Resampling methods

Confidence regions high-dimensional data nonasymptotic error control resampling cross-validation concentration inequalities resampled quantile


Arlot, Sylvain; Blanchard, Gilles; Roquain, Etienne. Some nonasymptotic results on resampling in high dimension, I: Confidence regions. Ann. Statist. 38 (2010), no. 1, 51--82. doi:10.1214/08-AOS667.

Export citation


  • [1] Arlot, S. (2007). Resampling and Model Selection. Ph.D. thesis, Univ. Paris XI.
  • [2] Arlot, S., Blanchard, G. and Roquain, É. (2010). Some nonasymptotic results on resampling in high dimension. II: Multiple tests. Ann. Statist. 38 83–99.
  • [3] Baraud, Y. (2004). Confidence balls in Gaussian regression. Ann. Statist. 32 528–551.
  • [4] Beran, R. (2003). The impact of the bootstrap on statistical algorithms and theory. Statist. Sci. 18 175–184.
  • [5] Beran, R. and Dümbgen, L. (1998). Modulation of estimators and confidence sets. Ann. Statist. 26 1826–1856.
  • [6] Cai, T. and Low, M. (2006). Adaptive confidence balls. Ann. Statist. 34 202–228.
  • [7] Cirel’son, B. R., Ibragimov, I. A. and Sudakov, V. N. (1976). Norms of Gaussian sample functions. In Proceedings of the Third Japan–USSR Symposium on Probability Theory. Lecture Notes in Mathematics 550 20–41. Springer, Berlin.
  • [8] Darvas, F., Rautiainen, M., Pantazis, D., Baillet, S., Benali, H., Mosher, J., Garnero, L. and Leahy, R. (2005). Investigations of dipole localization accuracy in MEG using the bootstrap. NeuroImage 25 355–368.
  • [9] DiCiccio, T. J. and Efron, B. (1996). Bootstrap confidence intervals. Statist. Sci. 11 189–228.
  • [10] Durot, C. and Rozenholc, Y. (2006). An adaptive test for zero mean. Math. Methods Statist. 15 26–60.
  • [11] Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Ann. Statist. 7 1–26.
  • [12] Fisher, R. A. (1935). The Design of Experiments. Oliver and Boyd, Edinburgh.
  • [13] Fromont, M. (2007). Model selection by bootstrap penalization for classification. Mach. Learn. 66 165–207.
  • [14] Ge, Y., Dudoit, S. and Speed, T. P. (2003). Resampling-based multiple testing for microarray data analysis. Test 12 1–77.
  • [15] Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.
  • [16] Hall, P. and Mammen, E. (1994). On general resampling algorithms and their performance in distribution estimation. Ann. Statist. 22 2011–2030.
  • [17] Hoffmann, M. and Lepski, O. (2002). Random rates in anisotropic regression. Ann. Statist. 30 325–396.
  • [18] Jerbi, K., Lachaux, J.-P., N’Diaye, K., Pantazis, D., Leahy, R. M., Garnero, L. and Baillet, S. (2007). Coherent neural representation of hand speed in humans revealed by MEG imaging. PNAS 104 7676–7681.
  • [19] Juditsky, A. and Lambert-Lacroix, S. (2003). Nonparametric confidence set estimation. Math. Methods Statist. 12 410–428.
  • [20] Koltchinskii, V. (2001). Rademacher penalties and structural risk minimization. IEEE Trans. Inform. Theory 47 1902–1914.
  • [21] Lepski, O. V. (1999). How to improve the accuracy of estimation. Math. Methods Statist. 8 441–486.
  • [22] Li, K.-C. (1989). Honest confidence regions for nonparametric regression. Ann. Statist. 17 1001–1008.
  • [23] Mason, D. M. and Newton, M. A. (1992). A rank statistics approach to the consistency of a general bootstrap. Ann. Statist. 20 1611–1624.
  • [24] Massart, P. (2007). Concentration Inequalities and Model Selection (Lecture Notes of the St-Flour Probability Summer School 2003). Lecture Notes in Mathematics 1896. Springer, Berlin.
  • [25] McDiarmid, C. (1989). On the method of bounded differences. In Surveys in Combinatorics. London Mathematical Society Lecture Notes 141 148–188. Cambridge Univ. Press, Cambridge.
  • [26] Pantazis, D., Nichols, T. E., Baillet, S. and Leahy, R. M. (2005). A comparison of random field theory and permutation methods for statistical analysis of MEG data. NeuroImage 25 383–394.
  • [27] Politis, D. N., Romano, J. P. and Wolf, M. (1999). Subsampling. Springer, New York.
  • [28] Præstgaard, J. and Wellner, J. A. (1993). Exchangeably weighted bootstraps of the general empirical process. Ann. Probab. 21 2053–2086.
  • [29] Robins, J. and van der Vaart, A. (2006). Adaptive nonparametric confidence sets. Ann. Statist. 34 229–253.
  • [30] Romano, J. P. and Wolf, M. (2005). Exact and approximate stepdown methods for multiple hypothesis testing. J. Amer. Statist. Assoc. 100 94–108.
  • [31] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York.
  • [32] Waberski, T., Gobbele, R., Kawohl, W., Cordes, C. and Buchner, H. (2003). Immediate cortical reorganization after local anesthetic block of the thumb: Source localization of somatosensory evoked potentials in human subjects. Neurosci. Lett. 347 151–154.