Renormalizing Experiments for Nonlinear Functionals

  • David L. Donoho


Let f = f(t), t R d be an unknown “object” (real-valued function), and suppose we are interested in recovering the nonlinear functional T(f). We know a priori that f ∈F, a certain convex class of functions (e.g. a class of smooth functions). For various types of measurements Yn=(yl, y2,…, yn), problems of this form arise in statistical settings, such as nonparametric density estimation and nonparametric regression estimation; but they also arise in signal recovery and image processing. In such problems, there generally exists an “optimal rate of convergence”: the minimax risk from n observations, \( R\left( n \right) = \mathop{{\inf }}\limits_{{\hat{T}}} \mathop{{\sup }}\limits_{{f \in F}} E{{\left( {\hat{T}\left( {{{Y}_{n}}} \right) - T\left( f \right)} \right)}^{2}}\) tends to zero as. \( R\left( n \right) \asymp {{n}^{{ - r}}}\) There is ariety of functionals T, function classes.F, and types of observation Yn; the literature is really too extensive to list here, although we mention Ibragimov & Has’minskii (1981), Sacks & Ylvisaker (1981), and Stone (1980). Lucien Le Cam (1973) has contributed directly to this literature, in his typical abstract and profound way; his ideas have stimulated the work of others in the field, e.g. Donoho & Liu (1991a).


Nonparametric Regression Unimodal Function Nonparametric Density Estimation Minimax Risk White Noise Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Brown, L. D. & Low, M. G. (1992), Asymptotic equivalence of nonparametric regression and white noise, Technical report, Department of Mathematics, Cornell University.Google Scholar
  2. Donoho, D. L. & Liu, R. C. (1991a), ‘Geometrizing rates of convergence, II’, Annals of Statistics 19 633–667.MathSciNetMATHCrossRefGoogle Scholar
  3. Donoho, D. L. & Liu, R. C. (1991b), ‘Geometrizing rates of convergence, III’, Annals of Statistics 19 668–701.MathSciNetMATHCrossRefGoogle Scholar
  4. Donoho, D. L. & Low, M. (1992), ‘Renormalization exponents and optimal pointwise rates of convergence’, Annals of Statistics 20 944–970.MathSciNetMATHCrossRefGoogle Scholar
  5. Donoho, D. L. & Nussbaum, M. (1990), ‘Minimax quadratic estimation of a quadratic functional’, Journal of Complexity 6 290–323.MathSciNetMATHCrossRefGoogle Scholar
  6. Ibragimov, I. A. & Has’minskii, R. Z. (1980), ‘Estimates of the signal, its derivatives, and point of maximum for gaussian distributions’, Theory of Probability and Its Applications 25 703–720.CrossRefGoogle Scholar
  7. Ibragimov, I. A. & Has’minskii, R. Z. (1981), Statistical Estimation: Asymptotic Theory, Springer-Verlag, New York.MATHGoogle Scholar
  8. Khasminskii, R. Z. & Lebedev, V. S. (1990), ‘On the properties of parametric estimators for areas of a discontinuous image’, Problems of Control and Information Theory pp. 375–385.Google Scholar
  9. Korostelev, A. P. (1987), ‘Minimax estimation of a discontinuous signal’, Theory of Probability and Its Applications 32 796–799.MathSciNetGoogle Scholar
  10. Korostelev, A. P. & Tsybakov, A. B. (1993), Minimax Theory of Image Reconstruction, Vol. 82 of Lecture Notes in Statistics, Springer-Verlag, New York.CrossRefGoogle Scholar
  11. Le Cam, L. (1973), ‘Convergence of estimates under dimensionality restrictions’, Annals of Statistics 19 633–667.Google Scholar
  12. Le Cam, L. (1986), Asymptotic Methods in Statistical Decision Theory, Springer-Verlag, New York.MATHCrossRefGoogle Scholar
  13. Low, M. G. (1992), ‘Renormalization and white noise approximation for nonparametric functional estimation problems’, Annals of Statistics 20 545–554.MathSciNetMATHCrossRefGoogle Scholar
  14. Nussbaum, M. (1993), Asymptotic equivalence of density estimation and white noise, Technical report, Institute for Applied Stochastic Analysis, Berlin.Google Scholar
  15. Sacks, J. & Ylvisaker, N. D. (1981), ‘Asymptotically optimum kernels for density estimation at a point’, Annals of Statistics 9, 334–346.MathSciNetMATHCrossRefGoogle Scholar
  16. Stone, C. J. (1980), ‘Optimum rates of convergence for nonparametric estimators’, Annals of Statistics 8, 1348–1360.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • David L. Donoho
    • 1
  1. 1.Stanford and University of California at BerkeleyUSA

Personalised recommendations