The Annals of Statistics

Adaptive confidence bands

Christopher Genovese and Larry Wasserman

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We show that there do not exist adaptive confidence bands for curve estimation except under very restrictive assumptions. We propose instead to construct adaptive bands that cover a surrogate function f which is close to, but simpler than, f. The surrogate captures the significant features in f. We establish lower bounds on the width for any confidence band for f and construct a procedure that comes within a small constant factor of attaining the lower bound for finite-samples.

Article information

Ann. Statist., Volume 36, Number 2 (2008), 875-905.

First available in Project Euclid: 13 March 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G15: Tolerance and confidence regions 62G08: Nonparametric regression 62G20: Asymptotic properties

Nonparametric confidence bands nonparametric regression adaptive confidence sets lower bounds


Genovese, Christopher; Wasserman, Larry. Adaptive confidence bands. Ann. Statist. 36 (2008), no. 2, 875--905. doi:10.1214/07-AOS500.

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  • Andrews, D. W. K. and Guggenberger, P. (2007). Hybrid and size-corrected subsample methods. Discussion Paper 1606, Cowles Foundation, Yale Univ.
  • Baraud, Y. (2002). Non-asymptotic minimax rates of testing in signal detection. Bernoulli 8 577–606.
  • Baraud, Y. (2004). Confidence balls in Gaussian regression. Ann. Statist. 32 528–551.
  • Beran, R. and Dümbgen, L. (1998). Modulation of estimators and confidence sets. Ann. Statist. 26 1826–1856.
  • Bickel, P. J. and Ritov, Y. (2000). Non- and semiparametric statistics: Compared and contrasted. J. Statist. Plann. Inference 91 209–228.
  • Birgé, L. (2001). An alternative point of view on Lepski’s method. In State of the Art in Probability and Statistics (M. de Gunst, C. Klaassen and A. van der Vaart, eds.) 113–133. IMS, Beachwood, OH.
  • Cai, T. and Low, M. (2006). Adaptive confidence balls. Ann. Statist. 34 202–228.
  • Cai, T. and Low, M. G. (2004). An adaptation theory for nonparametric confidence intervals. Ann. Statist. 32 1805–1840.
  • Chaudhuri, P. and Marron, J. S. (2000). Scale space view of curve estimation. Ann. Statist. 28 408–428.
  • Claeskens, G. and Van Keilegom, I. (2003). Bootstrap confidence bands for regression curves and their derivatives. Ann. Statist. 31 1852–1884.
  • Cummins, D., Filloon, T. and Nychka, D. (2001). Confidence intervals for nonparametric curve estimates: Toward more uniform pointwise coverage. J. Amer. Statist. Assoc. 96 233–246.
  • Davies, P. L. and Kovac, A. (2001). Local extremes, runs, strings and multiresolution (with discussion). Ann. Statist. 29 1–65.
  • Donoho, D. (1988). One-sided inference about functionals of a density. Ann. Statist. 16 1390–1420.
  • Donoho, D. (1995). De-noising by soft-thresholding. IEEE Trans. Inform. Theory 41 613–627.
  • Donoho, D. and Liu, R. (1991). Geometrizing rates of convergence. II. Ann. Statist. 19 633–667.
  • Donoho, D., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1995). Wavelet shrinkage: Asymptopia. J. Roy. Statist. Soc. Ser. B 57 301–369.
  • Eubank, R. L. and Speckman, P. L. (1993). Confidence bands in nonparametric regression. J. Amer. Statist. Assoc. 88 1287–1301.
  • Genovese, C. and Wasserman, L. (2005). Nonparametric confidence sets for wavelet regression. Ann. Statist. 33 698–729.
  • Hall, P. and Titterington, M. (1988). On confidence bands in nonparametric density estimation and regression. J. Multivariate Anal. 27 228–254.
  • Härdle, W. and Bowman, A. W. (1988). Bootstrapping in nonparametric regression: Local adaptive smoothing and confidence bands. J. Amer. Statist. Assoc. 83 102–110.
  • Härdle, W. and Marron, J. S. (1991). Bootstrap simultaneous error bars for nonparametric regression. Ann. Statist. 19 778–796.
  • Hoffman, O. and Lepski, M. (2002). Random rates in anisotropic regression (with discussion). Ann. Statist. 30 325–396.
  • Ingster, Y. (1993). Asymptotically minimax hypothesis testing for nonparametric alternatives, I and II. Math. Methods Statist. 2 85–114.
  • Ingster, Y. and Suslina, I. (2003). Nonparametric Goodness of Fit Testing Under Gaussian Models. Springer, New York.
  • Juditsky, A. and Lambert-Lacroix, S. (2003). Nonparametric confidence set estimation. Math. Methods Statist. 19 410–428.
  • Kabaila, P. (1995). The effects of model selection on confidence regions and prediction regions. Econometric Theory 11 537–549.
  • Kabaila, P. (1998). Valid confidence intervals in regression after variable selection. Econometric Theory 14 463–482.
  • Leeb, H. and Pötscher, B. M. (2005). Model selection and inference: Facts and fiction. Econometric Theory 21 21–59.
  • Li, K.-C. (1989). Honest confidence regions for nonparametric regression. Ann. Statist. 17 1001–1008.
  • Low, M. G. (1997). On nonparametric confidence intervals. Ann. Statist. 25 2547–2554.
  • Neumann, M. H. and Polzehl, J. (1998). Simultaneous bootstrap confidence bands in nonparametric regression. J. Nonparametr. Statist. 9 307–333.
  • Robins, J. and van der Vaart, A. (2006). Adaptive nonparametric confidence sets. Ann. Statist. 34 229–253.
  • Ruppert, D., Wand, M. P. and Carroll, R. J. (2003). Semiparametric Regression. Cambridge Univ. Press.
  • Sun, J. and Loader, C. R. (1994). Simultaneous confidence bands for linear regression and smoothing. Ann. Statist. 22 1328–1345.
  • Terrell, G. R. and Scott, D. W. (1985). Oversmoothed nonparametric density estimates. J. Amer. Statist. Assoc. 80 209–214.
  • Terrell, G. R. (1990). The maximal smoothing principle in density estimation. J. Amer. Statist. Assoc. 85 470–477.
  • Wahba, G. (1983). Bayesian “confidence intervals” for the cross-validated smoothing spline. J. Roy. Statist. Soc. Ser. B Methodol. 45 133–150.
  • Xia, Y. (1998). Bias-corrected confidence bands in nonparametric regression. J. Roy. Statist. Soc. Ser. B 60 797–811.