Estimating functionals, II

  • Arkadi Nemirovski
Topics In Non-parametric Statistics
Part of the Lecture Notes in Mathematics book series (LNM, volume 1738)


Unbiased Estimator Taylor Polynomial Wavelet Shrinkage Problem Inform Unknown Smoothness 
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  1. [1]
    Barron A. Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. on Information Theory, v. 39 No. 3 (1993).Google Scholar
  2. [2]
    Besov O.V., V.P. Il'in, and S.M. Nikol'ski. Integral representations of functions and embedding theorems, Moscow: Nauka Publishers, 1975) (in Russian)Google Scholar
  3. [3]
    Birgé L. Approximation dans les espaces métriques et théorie de l'estimation. Z. Wahrscheinlichkeitstheorie verw. Geb., v. 65 (1983), 181–237.CrossRefzbMATHGoogle Scholar
  4. [4]
    Cramer H. Mathematical Methods of Statistics, Princeton Univ. Press, Princeton, 1957.zbMATHGoogle Scholar
  5. [5]
    Donoho D., I. Johnstone. Ideal spatial adaptation via wavelet shrinkage. Biometrika v. 81 (1994) No.3, 425–455.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Donoho D., I. Johnstone. Adapting to unknown smoothness via wavelet shrinkage. J. Amer. Statist. Assoc. v. 90 (1995) No. 432, 1200–1224.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Donoho D., I. Johnstone, G. Kerkyacharian, D. Picard. Wavelet shrinkage: Asymptopia? (with discussion and reply by the authors). J. Royal Statist. Soc. Series B v. 57 (1995) No.2, 301–369.MathSciNetzbMATHGoogle Scholar
  8. [8]
    Eubank R. Spline smoothing and Nonparametric Regression, Dekker, New York, 1988.zbMATHGoogle Scholar
  9. [9]
    Goldenshluger A., A. Nemirovski. On spatially adaptive estimation of nonparametric regression. Math. Methods of Statistics, v. 6 (1997) No. 2, 135–170.MathSciNetzbMATHGoogle Scholar
  10. [10]
    Goldenshluger A., A. Nemirovski. Adaptive de-noising of signals satisfying differential inequalities. IEEE Transactions on Information Theory v. 43 (1997).Google Scholar
  11. [11]
    Golubev Yu. Asymptotic minimax estimation of regression function in additive model. Problemy peredachi informatsii, v. 28 (1992) No. 2, 3–15. (English transl. in Problems Inform. Transmission v. 28, 1992.)MathSciNetzbMATHGoogle Scholar
  12. [12]
    Härdle W., Applied Nonparametric Regression, ES Monograph Series 19, Cambridge, U.K., Cambridge University Press, 1990.CrossRefzbMATHGoogle Scholar
  13. [13]
    Ibragimov I.A., R.Z. Khasminski. Statistical Estimation: Asymptotic Theory, Springer, 1981.Google Scholar
  14. [14]
    Ibragimov I., A. Nemirovski, R. Khas'minski. Some problems of nonparametric estimation in Gaussian white noise. Theory Probab. Appl. v. 31 (1986) No. 3, 391–406.MathSciNetCrossRefGoogle Scholar
  15. [15]
    Juditsky, A. Wavelet estimators: Adapting to unknown smoothness. Math. Methods of Statistics v. 6 (1997) No. 1, 1–25.MathSciNetzbMATHGoogle Scholar
  16. [16]
    Juditsky A., A. Nemirovski. Functional aggregation for nonparametric estimation. Technical report # 993 (March 1996), IRISA, RennesGoogle Scholar
  17. [17]
    Korostelev A., A. Tsybakov. Minimax theory of image reconstruction. Lecture Notes in Statistics v. 82, Springer, New York, 1993.zbMATHGoogle Scholar
  18. [18]
    Lepskii O. On a problem of adaptive estimation in Gaussian white noise. Theory of Probability and Its Applications, v. 35 (1990) No. 3, 454–466.MathSciNetCrossRefGoogle Scholar
  19. [19]
    Lepskii O. Asymptotically minimax adaptive estimation I: Upper bounds. Optimally adaptive estimates. Theory of Probability and Its Applications, v. 36 (1991) No. 4, 682–697.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Lepskii O., E. Mammen, V. Spokoiny. Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimates with variable bandwidth selectors. Ann. Statist. v. 25 (1997) No.3, 929–947.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Nemirovski A., D. Yudin. Problem complexity and method efficiency in Optimization, J. Wiley & Sons, 1983.Google Scholar
  22. [22]
    Nemirovski A. On forecast under uncertainty. Problemy peredachi informatsii, v. 17 (1981) No. 4, 73–83. (English transl. in Problems Inform. Transmission v. 17, 1981.)MathSciNetGoogle Scholar
  23. [23]
    Nemirovski A. On nonparametric estimation of smooth regression functions. Sov. J. Comput. Syst. Sci., v. 23 (1985) No. 6, 1–11.MathSciNetGoogle Scholar
  24. [24]
    Nemirovski A. On necessary conditions for efficient estimation of functionals of a nonparametric signal in white noise. Theory Probab. Appl. v. 35 (1990) No. 1, 94–103.MathSciNetCrossRefGoogle Scholar
  25. [25]
    Nemirovski A. On nonparametric estimation of functions satisfying differential inequalities. — In: R. Khasminski, Ed. Advances in Soviet Mathematics, v. 12, American Mathematical Society, 1992, 7–43.Google Scholar
  26. [26]
    Pinsker M., Optimal filtration of square-integrable signals in Gaussian noise. Problemy peredachi informatsii, v. 16 (1980) No. 2, 120–133. (English transl. in Problems Inform. Transmission v. 16, 1980.)MathSciNetzbMATHGoogle Scholar
  27. [27]
    Pinsker M., S. Efroimovich. Learning algorithm for nonparametric filtering. Automation and Remote Control, v. 45 (1984) No. 11, 1434–1440.MathSciNetzbMATHGoogle Scholar
  28. [28]
    Pisier G. Remarques sur un resultat non publie de B. Maurey, — in: Seminaire d'analyse fonctionelle 1980–1981, v. 1–v. 12, Ecole Polytechnique, Palaiseau, 1981.Google Scholar
  29. [29]
    Prakasa Rao B.L.S. Nonparametric functional estimation. Academic Press, Orlando, 1983.zbMATHGoogle Scholar
  30. [30]
    Rosenblatt M. Stochastic curve estimation. Institute of Mathematical Statistics, Hayward, California, 1991.zbMATHGoogle Scholar
  31. [31]
    Suetin P.K. The classical orthogonal polynomials. Nauka, Moscow, 1976 (in Russian).zbMATHGoogle Scholar
  32. [32]
    Wahba G. Spline models for observational data. SIAM, Philadelphia, 1990.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2000

Authors and Affiliations

  • Arkadi Nemirovski
    • 1
  1. 1.Faculty of Industrial Engineering and Management, TechnionIsrael Institute of TechnologyTechnion City, HaifaIsrael

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