Asymptotically minimax estimation of a function with jumps

Catharina G.M. Oudshoorn

Full-text: Open access


Asymptotically minimax nonparametric estimation of a regression function observed in white Gaussian noise over a bounded interval is considered, with respect to a L2-loss function. The unknown function f is assumed to be m times differentiable except for an unknown although finite number of jumps, with piecewise mth derivative bounded in L2 norm. An estimator is constructed, attaining the same optimal risk bound, known as Pinsker's constant, as in the case of smooth functions (without jumps).

Article information

Bernoulli, Volume 4, Number 1 (1998), 15-33.

First available in Project Euclid: 6 April 2007

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

jump-point estimation nonparametric regression optimal constant tapered orthogonal series estimator


Oudshoorn, Catharina G.M. Asymptotically minimax estimation of a function with jumps. Bernoulli 4 (1998), no. 1, 15--33.

Export citation


  • [1] Belitser, E. and Levit, B. (1996) Asymptotically minimax nonparametric regression in L2. Statistics, 28, 105-122.
  • [2] Birgé, L. (1983) Approximation dans les espaces métriques et théorie de l'estimation. Z. Wahrscheinlichkeitstheorie Verw. Geb., 65, 181-237.
  • [3] Brown, L. and Low, M. (1996) Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist., 24, 2384-2398.
  • [4] Coddington, E. and Levinson, N. (1955) Theory of Ordinary Differential Equations. New York: McGraw-Hill.
  • [5] Donoho, D. (1994) Asymptotic risk for sup-norm loss: solution via optimal recovery. Probab. Theory Related Fields, 99, 145-170.
  • [6] Duistermaat, J. (1995) The Sturm-Liouville problem for the operator (-d²=dx²)^m, with Neumann or Dirichlet boundary conditions. Technical Report 899, Department of Mathematics, University of Utrecht.
  • [7] Dunford, N. and Schwartz, J. (1971) Linear operators, Part III: Spectral Operators. New York: Wiley Interscience.
  • [8] Efroimovich, S.Y. (1994) On nonparametric regression for i i d observations in general setting. Technical Report, University of New Mexico.
  • [9] Golubev, Y.K., Levit, B. and Tsybakov, A. (1995) Asymptotically efficient estimation of analytic functions in Gaussian noise. Technical Report 894, Department of Mathematics, University of Utrecht.
  • [10] Golubev, Y.K. and Nussbaum, M. (1990) A risk bound in Solobev class regression. Ann. Statist., 18, 758-778.
  • [11] Hall, P. and Patil, P. (1995) Formulae for mean integrated squared error of nonlinear wavelet-based density estimators. Ann. Statist., 23, 905-928.
  • [12] Ibragimov, I. and Hasminskii, R. (1981) Statistical Estimation: Asymptotic Theory. Berlin: Springer- Verlag.
  • [13] Korostelev, A. (1987) On minimax esimation of a discontinuous signal. Theory Probab. Appl., 32, 727-730.
  • [14] Korostelev, A. (1994) Exact asymptotic minimax estimate of a nonparametric regression in the uniform norm. Theory Probab. Appl., 38, 737-743.
  • [15] Mü ller, H. (1992) Change-points in nonparametric regression analysis. Ann. Statist., 20, 737-761.
  • [16] Neumark, M. (1967) Lineare differential Operatoren. Berlin: Akademie-Verlag.
  • [17] Nussbaum, M. (1985) Spline smoothing in regression models and asymptotic efficiency in L2. Ann. Statist., 13, 984-997.
  • [18] Pinsker, M. (1980) Optimal filtration of square-integrable signals of Gaussian noise. Prob. Inf. Trans., 16, 120-133.
  • [19] Speckman, P. (1985) Spline smoothing and optimal rates of convergence in nonparametric regression models. Ann. Statist., 13, 970-983.
  • [20] Stone, C. (1982) Optimal global rates of convergence for nonparametric regression. Ann. Statist., 10, 1040-1053.
  • [21] Wang, Y. (1995) Jump and sharp cusp detection by wavelets. Biometrika, 82, 385-397.