• Bernoulli
  • Volume 10, Number 5 (2004), 873-888.

Asymptotically exact minimax estimation in sup-norm for anisotropic Hölder classes

Karine Bertin

Full-text: Open access


We consider the Gaussian white noise model and study the estimation of a function f in the uniform norm assuming that f belongs to a Hölder anisotropic class. We give the minimax rate of convergence over this class and determine the minimax exact constant and an asymptotically exact estimator.

Article information

Bernoulli, Volume 10, Number 5 (2004), 873-888.

First available in Project Euclid: 4 November 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

anisotropic Hölder class minimax exact constant uniform norm white noise model


Bertin, Karine. Asymptotically exact minimax estimation in sup-norm for anisotropic Hölder classes. Bernoulli 10 (2004), no. 5, 873--888. doi:10.3150/bj/1099579160.

Export citation


  • [1] Adler, R.J. (1990) An Introduction to Continuity, Extrema, and Related Topics for general Gaussian Processes, IMS Lecture Notes - Monogr. Ser. 12. Hayward, CA: Institute of Mathematical Statistics.
  • [2] Barron, A., Birgé, L. and Massart, P. (1999) Risk bounds for model selection via penalization. Probab. Theory Related Fields, 113(3), 301-413.
  • [3] Bertin, K. (2004) Minimax exact constant in sup-norm for nonparametric regression with random design. J. Statist. Plann. Inference. To appear.
  • [4] Donoho, D.L. (1994) Asymptotic minimax risk for sup-norm loss: solution via optimal recovery. Probab. Theory Related Fields, 99(2), 145-170. Abstract can also be found in the ISI/STMA publication
  • [5] Gihman, I.I. and Skorohod, A.V. (1974) The Theory of Stochastic Processes. I. New York: Springer- Verlag.
  • [6] Gradshteyn, I.S. and Ryzhik, I.M. (1965) Table of Integrals, Series, and Products, Fourth edition prepared by Ju. V. Geronimus and M. Ju Ceîvtlin. Translated from the Russian by Scripta Technica, Inc. Translation edited by Alan Jeffrey, Academic Press, New York.
  • [7] Ibragimov, I.A. and Has´minskii, R.Z. (1981) Statistical Estimation. Asymptotic Theory. New York: Springer-Verlag.
  • [8] Kerkyacharian, G., Kepski, O. and Picard, D. (2001) Nonlinear estimation in anisotropic multi-index denoising. Probab. Theory Related Fields, 121(2), 137-170. Abstract can also be found in the ISI/STMA publication
  • [9] Klemelä, J. and Tsybakov, A.B. (2001) Sharp adaptive estimation of linear functionals. Ann. Statist., 29(6), 1567-1600.
  • [10] Korostelev, A.P. (1993) An asymptotically minimax regression estimator in the uniform norm up to a constant. Teor. Veroyatnost. i. Primenen., 38(4), 857-882. Abstract can also be found in the ISI/STMA publication
  • [11] Korostelev, A. and Nussbaum, M. (1999) The asyptotic minimax constant for sup-norm loss in nonparametric density estimation. Bernoulli, 5(6), 1099-1118.
  • [12] Lepski, O.V. and Tsybakov, A.B. (2000) Asymptotically exact nonparametric hypothesis testing in supnorm and at a fixed point. Probab. Theory Related Fields, 117(1), 17-48. Abstract can also be found in the ISI/STMA publication
  • [13] Lepskii, O.V. (1992) On problems of adpative estimation in white Gaussian noise. In R.Z. Khasminskij (ed.), Topics in Nonparametric Estimation, Adv. Soviet Math. 12, pp. 87-106. Providence, RI: American Mathematical Society.
  • [14] Lifshits, M.A. (1995) Gaussian Random Functions. Dordrecht: Kluwer Academic.
  • [15] Neumann, M.H. and von Sachs, R. (1997) Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra. Ann. Statist., 25(1), 38-76. Abstract can also be found in the ISI/STMA publication
  • [16] Nussbaum, M. (1986) On the nonparametric estimation of regression functions that are smooth in a domain in Rk. Teor. Veroyatnost. i Primenen., 31(1), 118-125.
  • [17] Piterbarg, V.I. (1996) Asymptotic Methods in the Theory of Gaussian Processes and Fields, Transl. Math. Monogr. 148. Providence, RI: American Mathematical Society.
  • [18] Stone, C.J. (1982) Optimal global rates of convergence for nonparametric regression. Ann. Statist., 10(4), 1040-1053.