Skip to main content
SpringerLink
Log in

Estimation and detection of high-variable functions from Sloan—Woźniakowski space

  • Published:
Mathematical Methods of Statistics Aims and scope Submit manuscript

Abstract

The major difficulty arising in statistics of multi-variable functions is “the curse of dimensionality”: the rates of accuracy in estimation and separation rates in detection problems behave poorly when the number of variables increases. This difficulty arises for most popular functional classes such as Sobolev or Hölder balls.

In this paper we consider functional classes of a new type, first introduced by Sloan and Woźniakowski in 1998. We consider balls F σ,s in a “Sloan—Woźniakowski” or “weighted Sobolev” space characterized by two parameters: σ > 0 is a “smoothness” parameter, and s > 0 determines the weight sequence which describes “importance” of the variables. Previously Kuo and Sloan [18] used the spaces of similar structure to address the problem of numerical integration.

For the classes F σ,s we show that in the white Gaussian noise model, the separation rates in detection are similar to those for one-variable functions of smoothness σ* = min(s,σ) regardless of the original problem dimension; thus the curse of dimensionality is “lifted”. Similar results hold for the estimation problem.

The studies are based on known results for estimation and detection problems for ellipsoids. Using these results, the asymptotics in the problems are determined by asymptotics of “distribution of coefficients” of ellipsoids. The key point of the paper is the study of these asymptotics for the balls F σ,s .

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. E. N. Belitser and B. Y. Levit, “On Minimax Filtering over Ellipsoids”, Math. Methods Statist. 4, 259–273 (1995).

    MATH  MathSciNet  Google Scholar 

  2. R. Bellman, Dynamic Programming (Princeton Univ. Press, Princeton—New York, 1957).

    Google Scholar 

  3. L. D. Brown and M. G. Low, “Asymptotic Equivalence of Nonparametric Regression and White Noise”, Ann. Statist. 24, 2384–2398 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  4. T. Gai and M. G. Low, “Nonparametric Estimation over Shrinking Neighborhoods: Superefficiency and Adaptation”, Ann. Statist. 33, 184–213 (2002).

    Google Scholar 

  5. D. L. Donoho and I. M. Johnstone, “Minimax Estimation via Wavelet Shrinkage”, Ann. Statist. 26, 668–701 (1998).

    MathSciNet  Google Scholar 

  6. D. L. Donoho, “High-Dimensional Data Analysis: The Curses and Blessings of Dimensionality”, in Lecture on the AMS Conference “Mathematical Challenges of the 21st Century”, Los Angeles, August 6–11, 2000. http://www-stat.stanford.edu/ donoho/Lectures/AMS2000/AMS2000.html

  7. S. V. Efromovich, Nonparametric Curve Estimation (Springer, New York, 1999).

    MATH  Google Scholar 

  8. M. S. Ermakov, “Minimax Detection of a Signal in a Gaussian White Noise”, Theory Probab. Appl. 35, 667–679 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  9. I. A. Ibragimov and R. Z. Khasminskii, “One Problem of Statistical Estimation in a White Gaussian Noise”, Soviet Math. Dokl. 236(4), 333–337 (1977).

    Google Scholar 

  10. I. A. Ibragimov and R. Z. Khasminskii, “Some Estimation Problems on Infinite Dimensional Gaussian White Noise”, in Festschrift for Lucien Le Cam. Research papers in Probab. and Statist. (Springer, New York, 1997), pp. 275–296.

    Google Scholar 

  11. Yu. I. Ingster, “Minimax Nonparametric Detection of Signals in White Gaussian Noise”, Problems Inform. Transmission 18, 130–140 (1982).

    MATH  MathSciNet  Google Scholar 

  12. Yu. I. Ingster, “Asymptotically Minimax Testing of Nonparametric Hypotheses”, in Proc. 4th Vilnius Conference on Probab. Theory and Math. Statist. (VNU Science Press, Vilnius, 1987), Vol. 1, pp. 553–573.

    Google Scholar 

  13. Yu. I. Ingster, “Asymptotically Minimax Hypothesis Testing for Nonparametric Alternatives. I, II, III”, Math. Methods Statist. 2, 85–114, 171–189, 249–268 (1993).

    MATH  MathSciNet  Google Scholar 

  14. Yu. I. Ingster and I. A. Suslina, Nonparametric Goodness-of-Fit Testing under Gaussian Model, in Lectures Notes in Statist. (Springer, New York, 2002), Vol. 169.

    Google Scholar 

  15. Yu. I. Ingster and I. A. Suslina, “On Estimation and Detection of Smooth Functions of Many Variables”, Math. Methods Statist. 14(3), 299–331 (2005).

    MathSciNet  Google Scholar 

  16. Yu. I. Ingster and I. A. Suslina, “On Estimation and Detection of Infinite-Variable Function”, Zapiski Nauchn. Seminar. POMI 328 (2006) [J. Math. Sci. 139 (3), 6548–6561 (2006)].

  17. G. Kerkyacharian, O. Lepski, and D. Picard, “Nonlinear Estimation and Anisotropic Multi-Index Denoising”, Probab. Theory and Rel. Fields 121, 137–170 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  18. F. Y. Kuo and J. H. Sloan, “Lifting the Curse of Dimensionality”, Notices of AMS 52(11), 1320–1329 (2005).

    MATH  MathSciNet  Google Scholar 

  19. Y. Lin, “Tensor Product Space ANOVA Model”, Ann. Statist. 28(3), 734–755 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  20. M. Nussbaum, “Asymptotic Equivalence of Density Estimation and Gaussian White Noise”, Ann. Statist. 24, 2399–2430 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  21. V. V. Petrov, Sums of Independent Random Variables (Springer, New York, 1975).

    Google Scholar 

  22. M. S. Pinsker, “Optimal Filtration of Square-Integrable Signals in Gaussian Noise”, Problems Inform. Transmission 16(2), 120–133 (1980).

    MATH  MathSciNet  Google Scholar 

  23. A. V. Skorohod, Integration in Hilbert Spaces (Springer, Berlin—New York, 1974).

    Google Scholar 

  24. I. H. Sloan and H. Woźniakowski, “When are Quasi-Monte Carlo Algorithms Efficient for High-Dimensional Integrals?” J. Complexity 14, 1–33 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  25. Ch. Stone, “Additive Regression and Other Nonparamtric Models”, Ann. Statist. 13(2), 689–705 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  26. H. Woźniakowski, “Tractability of Multivariate Problems for Weighted Spaces of Functions”, in Approximation and Probability (Banach Center Publications, Inst. of Math., Polish Acad. of Sci., Warszawa, 2006), Vol. 72.

    Google Scholar 

  27. Yu. I. Ingster and Th. Sapatinas, Minimax Goodness-of-Fit Gesting in Multivariate Nonparametric Regression, Rechnical Report TR-22-2007 (Univ. of Cyprus, 2007). http://www.mas.ucy.ac.cy/english/technical reports eng207.htm

Download references

Author information

Authors and Affiliations

  1. St. Petersburg State Electrotechnical University, St. Petersburg, Russia

    Yu. Ingster

  2. State University Information Technologies, Mechanics and Optics, St. Petersburg, Russia

    I. Suslina

Authors
  1. Yu. Ingster
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. Suslina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu. Ingster.

About this article

Cite this article

Ingster, Y., Suslina, I. Estimation and detection of high-variable functions from Sloan—Woźniakowski space. Math. Meth. Stat. 16, 318–353 (2007). https://doi.org/10.3103/S1066530707040035

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066530707040035

Key words

2000 Mathematics Subject Classification

Search

Navigation