Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Structural adaptation via \(\mathbb{L}_p\) -norm oracle inequalities
Download PDF
Download PDF
  • Published: 30 November 2007

Structural adaptation via \(\mathbb{L}_p\) -norm oracle inequalities

  • Alexander Goldenshluger1 &
  • Oleg Lepski2 

Probability Theory and Related Fields volume 143, pages 41–71 (2009)Cite this article

  • 188 Accesses

  • 32 Citations

  • Metrics details

Abstract

In this paper we study the problem of adaptive estimation of a multivariate function satisfying some structural assumption. We propose a novel estimation procedure that adapts simultaneously to unknown structure and smoothness of the underlying function. The problem of structural adaptation is stated as the problem of selection from a given collection of estimators. We develop a general selection rule and establish for it global oracle inequalities under arbitrary \({\mathbb{L}}_p\) -losses. These results are applied for adaptive estimation in the additive multi-index model.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Barron, A., Birgé, L., Massart, P.: Risk bounds for model selection via penalization. Probab. Theory Relat. Fields 113, 301–413 (1999)

    Article  MATH  Google Scholar 

  2. Belomestny, D., Spokoiny, V.: Local likelihood modeling via stagewise aggregation. WIAS preprint No. 1000 (2004). www.wias-berlin.de

  3. Bertin, K.: Asymptotically exact minimax estimation in sup-norm for anisotropic Hölder balls. Bernoulli 10, 873–888 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cavalier, L., Golubev, G.K., Picard, D., Tsybakov, A.B.: Oracle inequalities for inverse problems. Ann. Stat. 30, 843–874 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen, H.: Estimation of a projection-pursuit type regression model. Ann. Stat. 19, 142–157 (1991)

    Article  MATH  Google Scholar 

  6. Devroye, L., Lugosi, G.: Combinatorial Methods in Density Estimation. Springer, New York (2001)

    MATH  Google Scholar 

  7. Folland, G.B.: Real Analysis, 2nd edn. Wiley, New York (1999)

    MATH  Google Scholar 

  8. Goldenshluger, A., Nemirovski, A.: On spatially adaptive estimation of nonparametric regression. Math. Methods Stat. 6, 135–170 (1997)

    MATH  MathSciNet  Google Scholar 

  9. Golubev, G.K.: Asymptotically minimax estimation of a regression function in an additive model. Probl. Inform. Transm. 28, 101–112 (1992)

    MathSciNet  Google Scholar 

  10. Golubev, G.K.: The method of risk envelopes in the estimation of linear functionals (Russian). Probl. Inform. Transm. 40, 53–65 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  11. Györfi, L., Kohler, M., Krzyzak, A., Walk, H.: A Distribution-Free Theory of Nonparametric Regression. Springer, New York (2002)

    MATH  Google Scholar 

  12. Hall, P.: On projection-pursuit regression. Ann. Stat. 17, 573–588 (1989)

    Article  MATH  Google Scholar 

  13. Hristache, M., Juditsky, A., Spokoiny, V.: Direct estimation of the index coefficient in a single-index model. Ann. Stat. 29, 595–623 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hristache, M., Juditsky, A., Polzehl, J., Spokoiny, V.: Structure adaptive approach for dimension reduction. Ann. Stat. 29, 1537–1566 (2001)

    MATH  MathSciNet  Google Scholar 

  15. Huber, P.: Projection pursuit. With discussion. Ann. Stat. 13, 435–525 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ibragimov, I.A., Khasminskii, R.Z.: Bounds for the quality of nonparametric estimation of regression. Theory Probab. Appl. 27, 81–94 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ibragimov, I.A.: Estimation of multivariate regression. Theory Probab. Appl. 48, 256–272 (2004)

    Article  MathSciNet  Google Scholar 

  18. Jennrich, R.: Asymptotic properties of non-linear least squares estimators. Ann. Math. Stat. 40, 633–643 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  19. Johnstone, I.M.: Oracle inequalities and nonparametric function estimation. In: Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998). Doc. Math., Extra vol. III, pp. 267–278 (1998)

  20. Juditsky, A., Lepski, O., Tsybakov, A.: Statistical estimation of composite functions. Manuscript (2006)

  21. Kerkyacharian, G., Lepski, O., Picard, D.: Nonlinear estimation in anisotropic multi-index denoising. Probab. Theory Relat. Fields 121, 137–170 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  22. Lepski, O.V., Levit, B.Y.: Adaptive nonparametric estimation of smooth multivariate functions. Math. Methods Stat. 8, 344–370 (1999)

    MATH  MathSciNet  Google Scholar 

  23. Lepski, O., Mammen, E., Spokoiny, V.: Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimators with variable bandwidth selectors. Ann. Stat. 25, 929–947 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  24. Lepski, O.V., Spokoiny, V.G.: Optimal pointwise adaptive methods in nonparametric estimation. Ann. Stat. 25, 2512–2546 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  25. Lifshits, M.: Gaussian Random Functions. Kluwer, Dordrecht (1995)

    Google Scholar 

  26. Nemirovski, A.S.: Nonparametric estimation of smooth regression functions. Soviet J. Comput. Syst. Sci. 23(6), 1–11 (1985); translated from Izv. Akad. Nauk SSSR Tekhn. Kibernet. 235 (3), 50–60 (1985)(Russian)

  27. Nemirovski, A.: Topics in Non-parametric Statistics. Lectures on probability theory and statistics (Saint-Flour, 1998), Lecture Notes in Mathematics, vol. 1738, pp. 85–277. Springer, Berlin (2000)

  28. Nicoleris, T., Yatracos, Y.: Rates of convergence of estimators, Kolmogorov’s entropy and the dimensionality reduction principle in regression. Ann. Stat. 25, 2493–2511 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  29. Nussbaum, M.: Nonparametric estimation of a regression function that is smooth in a domain in \({\mathbb{R}}^k\) . Theory Probab. Appl. 31, 108–115 (1987)

    Article  MATH  Google Scholar 

  30. Stone, C.J.: Optimal global rates of convergence for nonparametric regression. Ann. Stat. 10, 1040–1053 (1982)

    Article  MATH  Google Scholar 

  31. Stone, C.J.: Additive regression and other nonparametric models. Ann. Stat. 13, 689–705 (1985)

    Article  MATH  Google Scholar 

  32. Talagrand, M.: Sharper bounds for Gaussian and empirical processes. Ann. Probab. 22, 28–76 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  33. Tsybakov, A.: Optimal rates of aggregation. In: Scholkopf, B., Warmuth, M. (eds) Computational Learning Theory and Kernel Machines. Lectures Notes in Artificial Intelligence, vol.2777, pp. 303–313. Springer, Heidelberg (2003)

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, University of Haifa, 31905, Haifa, Israel

    Alexander Goldenshluger

  2. Laboratoire d’Analyse, Topologie et Probabilités UMR CNRS 6632, Université de Provence, 39, rue F.Joliot Curie, 13453, Marseille, France

    Oleg Lepski

Authors
  1. Alexander Goldenshluger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Oleg Lepski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Goldenshluger.

Additional information

Supported by the ISF grant No. 389/07.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Goldenshluger, A., Lepski, O. Structural adaptation via \(\mathbb{L}_p\) -norm oracle inequalities. Probab. Theory Relat. Fields 143, 41–71 (2009). https://doi.org/10.1007/s00440-007-0119-5

Download citation

  • Received: 19 April 2007

  • Revised: 31 October 2007

  • Published: 30 November 2007

  • Issue Date: January 2009

  • DOI: https://doi.org/10.1007/s00440-007-0119-5

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Structural adaptation
  • Oracle inequalities
  • Minimax risk
  • Adaptive estimation
  • Optimal rates of convergence

Mathematics Subject Classification (2000)

  • 62G05
  • 62G20
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature