The Annals of Statistics

Confidence bands in density estimation

Evarist Giné and Richard Nickl

Full-text: Open access


Given a sample from some unknown continuous density f : ℝ→ℝ, we construct adaptive confidence bands that are honest for all densities in a “generic” subset of the union of t-Hölder balls, 0<tr, where r is a fixed but arbitrary integer. The exceptional (“nongeneric”) set of densities for which our results do not hold is shown to be nowhere dense in the relevant Hölder-norm topologies. In the course of the proofs we also obtain limit theorems for maxima of linear wavelet and kernel density estimators, which are of independent interest.

Article information

Ann. Statist., Volume 38, Number 2 (2010), 1122-1170.

First available in Project Euclid: 19 February 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation
Secondary: 60F05: Central limit and other weak theorems

Adaptive estimation limit theorem density estimation extremes Gaussian processes wavelet estimators kernel estimators


Giné, Evarist; Nickl, Richard. Confidence bands in density estimation. Ann. Statist. 38 (2010), no. 2, 1122--1170. doi:10.1214/09-AOS738.

Export citation


  • Banach, S. (1931). Über die Baire’sche Kategorie gewisser Funktionenmengen. Studia Math. 3 174–179.
  • 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. and Rosenblatt, M. (1973). On some global measures of the deviations of density function estimates. Ann. Statist. 1 1071–1095. Correction (1975) 3 1370.
  • Cai, T. T. and Low, M. G. (2004). An adaptation theory for nonparametric confidence intervals. Ann. Statist. 32 1805–1840.
  • Cai, T. T. and Low, M. G. (2006). Adaptive confidence balls. Ann. Statist. 34 202–228.
  • Daubechies, I. (1992). Ten Lectures on Wavelets. CBMS-NSF Reg. Conf. Ser. in Appl. Math. 61. SIAM, Philadelphia, PA.
  • Davies, P. L., Kovac, A. and Meise, M. (2009). Nonparametric regression, confidence regions and regularization. Ann. Statist. 37 2597–2625.
  • Davis, R. A. (1979). Maxima and minima of stationary sequences. Ann. Probab. 7 453–460.
  • de la Peña, V. and Giné, E. (1999). Decoupling: From Dependence to Independence. Springer, New York.
  • Dudley, R. M. (1985). An extended Wichura theorem, definitions of Donsker class, and weighted empirical distributions. In Probability in Banach Spaces, V (Medford, MA, 1984). Lecture Notes in Math. 1153 141–178. Springer, Berlin.
  • Dudley, R. M. (1999). Uniform Central Limit Theorems. Cambridge Univ. Press.
  • Dümbgen, L. (2003). Optimal confidence bands for shape-restricted curves. Bernoulli 9 423–449.
  • Genovese, C. and Wasserman, L. (2005). Nonparametric confidence sets for wavelet regression. Ann. Statist. 33 698–729.
  • Genovese, C. and Wasserman, L. (2008). Adaptive confidence bands. Ann. Statist. 36 875–905.
  • Giné, E. and Guillou, A. (2002). Rates of strong uniform consistency for multivariate kernel density estimators. Ann. Inst. H. Poincaré Probab. Statist. 38 907–921.
  • Giné, E., Koltchinskii, V. and Sakhanenko, L. (2004). Kernel density estimators: Convergence in distribution for weighted sup-norms. Probab. Theory Related Fields 130 167–198.
  • Giné, E. and Madych, W. R. (2009). On the periodized square of L2 cardinal splines. Preprint.
  • Giné, E. and Nickl, R. (2010). Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections. Bernoulli. To appear.
  • Giné, E. and Nickl, R. (2009a). An exponential inequality for the distribution function of the kernel density estimator, with applications to adaptive estimation. Probab. Theory Related Fields 143 569–596.
  • Giné, E. and Nickl, R. (2009b). Uniform limit theorems for wavelet density estimators. Ann. Probab. 37 1605–1646.
  • Goldenshluger, A. and Lepski, O. (2009). Structural adaptation via Lp-norm oracle inequalities. Probab. Theory Related Fields 143 41–71.
  • Hall, P. (1992). Effect of bias estimation on coverage accuracy of bootstrap confidence intervals for a probability density. Ann. Statist. 20 675–694.
  • Härdle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998). Wavelets, Approximation, and Statistical Applications. Lecture Notes in Statistics 129. Springer, New York.
  • Hengartner, N. W. and Stark, P. B. (1995). Finite-sample confidence envelopes for shape-restricted densities. Ann. Statist. 23 525–550.
  • Hoffmann, M. and Lepski, O. (2002). Random rates in anisotropic regression (with discussion). Ann. Statist. 30 325–396.
  • Hüsler, J. (1999). Extremes of Gaussian processes, on results of Piterbarg and Seleznjev. Statist. Probab. Lett. 44 251–258.
  • Hüsler, J., Piterbarg, V. and Seleznjev, O. (2003). On convergence of the uniform norms for Gaussian processes and linear approximation problems. Ann. Appl. Probab. 13 1615–1653.
  • Jaffard, S. (1997). Multifractal formalism for functions. I. Results valid for all functions. SIAM J. Math. Anal. 28 944–970.
  • Jaffard, S. (2000). On the Frisch–Parisi conjecture. J. Math. Pures Appl. 79 525–552.
  • Juditsky, A. and Lambert-Lacroix, S. (2003). Nonparametric confidence set estimation. Math. Methods Statist. 19 410–428.
  • Komlós, J., Major, J. and Tusnády, G. (1975). An approximation of partial sums of independent rv’s, and the sample df. I. Z. Wahrsch. Verw. Gebiete 32 111–131.
  • Konstant, D. G. and Piterbarg, V. I. (1993). Extreme values of the cyclostationary Gaussian random processes. J. Appl. Probab. 30 82–97.
  • Leadbetter, M., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
  • 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.
  • Meyer, Y. (1992). Wavelets and Operators I. Cambridge Univ. Press.
  • Nolan, D. and Pollard, D. (1987). U-processes: Rates of convergence. Ann. Statist. 15 780–799.
  • Picard, D. and Tribouley, K. (2000). Adaptive confidence interval for pointwise curve estimation. Ann. Statist. 28 298–335.
  • Piterbarg, V. I. and Seleznjev, O. (1994). Linear interpolation of random processes and extremes of a sequence of Gaussian nonstationary processes. Technical Report 1994, 446, Center Stoch. Process, North Carolina Univ., Chapel Hill.
  • Robins, J. and van der Vaart, A. W. (2006). Adaptive nonparametric confidence sets. Ann. Statist. 34 229–253.
  • Schumaker, L. L. (1993). Spline Functions: Basic Theory. Krieger, Malabar. (Correlated reprint of the 1981 original.)
  • Smirnov, N. V. (1950). On the construction of confidence regions for the density of distribution of random variables. Doklady Akad. Nauk SSSR 74 189–191 (Russian).
  • Talagrand, M. (1996). New concentration inequalities in product spaces. Invent. Math. 126 505–563.