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
Complete cubic spline estimation of non-parametric regression functions
Download PDF
Download PDF
  • Published: March 1990

Complete cubic spline estimation of non-parametric regression functions

  • Václav Fabian1 

Probability Theory and Related Fields volume 85, pages 57–64 (1990)Cite this article

  • 146 Accesses

  • 3 Citations

  • Metrics details

Summary

For regression functions on [0, 1] with bounded fourth derivatives, a complete cubic spline estimate is proposed and shown to have an asymptotically optimal error rate among all estimates. The error is measured by the supremum norm.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  • Chen, H.: Lower rate of convergence for locating a maximum of a function. Ann. Stat.16, 1330–1334 (1988)

    Google Scholar 

  • De Boor, C.: A practical guide to splines. Berlin Heidelberg New York: Springer 1978

    Google Scholar 

  • Fabian, V.: On asymptotic normality in stochastic approximation. Ann. Math. Stat.39, 1327–1332 (1968)

    Google Scholar 

  • Fabian, V.: Polynomial estimation of regression functions with the supremum norm error. Ann. Stat.16, 1345–1368 (1988)

    Google Scholar 

  • Fabian, V., Hannan, J.: Introduction to probability and mathematical statistics. New York: Wiley 1985

    Google Scholar 

  • Forsythe, G.E., Malcolm, M.A., Moler, C.B.: Computer methods for mathematical computation. Englewood Cliffs, NJ: Prentice-Hall 1977

    Google Scholar 

  • Halász, G.: Statistic interpolation. In: Alexits, G., Turan, P. (eds) Fourier analysis and approximation theory, vol. I, pp. 403–410. Amsterdam: North-Holland 1978

    Google Scholar 

  • Hall, C.A.: On error bounds for spline interpolation. J. Approx. Theory1, 209–218 (1968)

    Google Scholar 

  • Hall, C.A., Meyer, W.W.: Optimal error bounds for cubic spline interpolation. J. Approx. Theory16, 105–122 (1976)

    Google Scholar 

  • Koronacki, J.: Kernel estimation of smooth densities using Fabian's approach. Statistics18, 37–47 (1987)

    Google Scholar 

  • Ibragimov, I.A., Has'minskij, R.Z.: On nonparametric estimation of regression. Sov. Math., Dokl.21, 810–814 (1980)

    Google Scholar 

  • Ibragimov, I.A., Has'minskij, R.Z.: Asymptotic bounds on the quality of the nonparametric regression estimation inL p (russian). Zapiski Naucnych Seminarov LOMI97, 88–101 (1981). Translation: J. Sov. Math.24, 540–550 (1984)

    Google Scholar 

  • Ibragimov, I.A., Has'minskij, R.Z.: Bounds for the risks of nonparametric estimates of the regression. (russian) Teorija verojatn. i prim.32, 81–94 (1982). Translation: Theory Probab. Appl.27, 84–99

    Google Scholar 

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

    Google Scholar 

  • Wahba, G., Wold, S.: A completely automatic French curve: fitting spline functions by cross validátion. Commun. Stat.4, 1–17 (1975)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics and Probability, Michigan State University, 48824-1027, East Lansing, MI, USA

    Václav Fabian

Authors
  1. Václav Fabian
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Fabian, V. Complete cubic spline estimation of non-parametric regression functions. Probab. Th. Rel. Fields 85, 57–64 (1990). https://doi.org/10.1007/BF01377628

Download citation

  • Received: 02 August 1987

  • Revised: 04 September 1989

  • Issue Date: March 1990

  • DOI: https://doi.org/10.1007/BF01377628

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

  • Error Rate
  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Regression Function
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