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
On estimating a density using Hellinger distance and some other strange facts
Download PDF
Download PDF
  • Published: January 1986

On estimating a density using Hellinger distance and some other strange facts

  • Lucien Birgé1,2 

Probability Theory and Related Fields volume 71, pages 271–291 (1986)Cite this article

Résumé

On s'intéresse ici aux possibles vitesses d'estimation d'une densité à support compact dans ℝm sous des hypothèses de régularité, lorsque la perte est mesurée par le carré de la distance de Hellinger (on regardera aussi le cas connu des normes \(\mathbb{L}^q \) pour 1≦q≦2) et le risque est le risque minimax sur la famille. On donne une méthode générale permettant de traiter les problèmes dans le cadre de la théorie de l'approximation sous des conditions concernant l'entropie métrique et l' ε-capacité des familles à estimer. Les rapports entre régularité et entropie métrique étant bien connus, nous pourrons aussi traiter les cas classiques et d'autres qui le sont moins. Sous des conditions de bornes inférieures les vitesses sont celles observées pour la norme \(\mathbb{L}^q \) mais elles diffèrent dans le cas général. On montre aussi que les restrictions sur la compacité du support ou la régularité sont indispensables et que leur absence mène à l'impossibilité d'obtenir une estimation raisonnable en ce sens que n'importe quelle suite d'estimateurs sera arbitrairement mauvaise en un point au moins. Un résultat analogue est vrai sous des conditions de régularité.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Assouad, P.: Deux remarques sur l'estimation. C.R. Acad. Sci. Paris 296, Sér. I, 1021–1024 (1983)

    Google Scholar 

  2. Birgé, L.: Thèse, 3e-partie. Université Paris VII (1980)

  3. Birgés, L.: Approximation dans les espaces métriques et théorie de l'estimation. Z. Wahrscheinlichkeitstheor. Verw. Geb. 65, 181–237 (1983)

    Google Scholar 

  4. Birgé, L.: Non-asymptotic minimax risk for Hellinger balls. Probability and Math. Statistics. To appear

  5. Bretagnolle, J., Huber, C.: Estimation des densités: risque minimax. Z. Wahrscheinlichkeitstheor. Verw. Geb. 47, 119–137 (1979)

    Google Scholar 

  6. Dacunha-Castelle, D.: École d'Eté de Probabilités de Saint-Flour VII. Lecture Notes in Mathematics no 678. Berlin-Heidelberg-New York: Springer 1977

    Google Scholar 

  7. Devroye, L.: On arbitrarily slow rates of convergence in density estimation. Z. Wahrscheinlichkeitstheor. Verw. Gebiete 62, 475–483 (1983)

    Google Scholar 

  8. Ibragimov, I.A., Khas'minskii, R.Z.: Estimation of distribution density. Zap. Nauchn. Semin. LOMI 98, 61–85 (1980) in russian, and J. Sov. Math. 21, 40–57 (1983)

    Google Scholar 

  9. Ibragimov, I.A., Khas'minskii, R.Z.: On the non-parametric density function. Zap. Nauchn. Semin. LOMI 108, 73–89, in russian (1981)

    Google Scholar 

  10. Ibragimov, I.A., Khas'minskii, R.Z.: Statistical Estimation, Asymptotic Theory. Berlin-Heidelberg-New York: Springer 1981

    Google Scholar 

  11. Khas'minskii, R.Z.: A lower bound on the risks of non-parametric estimates of densities in the uniform metric. Theory Probab. Appl. 23, 794–796 (1978)

    Google Scholar 

  12. Kolmogorov, A.N., Tikhomirov, V.M.: ε-entropy and ε-capacity of sets in function spaces. Am. Math. Soc. Transl. (2) 17, 277–364 (1961)

    Google Scholar 

  13. Le Cam, L.: Convergence of estimates under dimensionality restrictions. Ann. Statist. 1, 38–53 (1973)

    Google Scholar 

  14. LeCam, L.: Asymptotic methods in statistical decision theory. To be published

  15. Lorentz, G.G.: Metric entropy and approximation. Bull. Amer. Math. Soc. 72, 903–937 (1966)

    Google Scholar 

  16. Lorentz, G.G.: Approximation of Functions. New York: Holt, Rinehart, Winston 1966

    Google Scholar 

  17. Müller, H.G., Gasser, T.: Optimal convergence properties of kernel estimates of derivatives of a density function. In: Smoothing Techniques for Curve Estimation (T. Gasser and M. Rosenblatt eds.). Lecture Notes in Mathematics no. 757, pp. 144–154. Berlin-Heidelberg-New York: Springer 1979

    Google Scholar 

  18. Stone, C.J.: Optimal rates of convergence for nonparametric estimators. Ann. Statist. 8, 1348–1360 (1980)

    Google Scholar 

  19. Stone, C.J.: Optimal uniform rate of convergence for nonparametric estimators of a density function or its derivatives. University of California Berkeley: preprint (1983)

Download references

Author information

Authors and Affiliations

  1. U.E.R. de Sciences Econoniques, Université Paris X-Nanterre, 200 Av. de la République, F-92001, Nanterre-Cedex, France

    Lucien Birgé

  2. Mathematical Sciences Research Institute, Berkeley, California, USA

    Lucien Birgé

Authors
  1. Lucien Birgé
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work was carried out during a visit of the author at the Mathematical Sciences Research Institute at Berkeley

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Birgé, L. On estimating a density using Hellinger distance and some other strange facts. Probab. Th. Rel. Fields 71, 271–291 (1986). https://doi.org/10.1007/BF00332312

Download citation

  • Received: 15 June 1983

  • Revised: 28 May 1985

  • Issue Date: January 1986

  • DOI: https://doi.org/10.1007/BF00332312

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

  • Stochastic Process
  • Probability Theory
  • Statistical Theory
  • Support Compact
  • Hellinger Distance
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