Skip to main content
SpringerLink
Log in

Optimal design for smoothing splines

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

In the common nonparametric regression model we consider the problem of constructing optimal designs, if the unknown curve is estimated by a smoothing spline. A special basis for the space of natural splines is introduced and the local minimax property for these splines is used to derive two optimality criteria for the construction of optimal designs. The first criterion determines the design for a most precise estimation of the coefficients in the spline representation and corresponds to D-optimality, while the second criterion is the G-optimality criterion and corresponds to an accurate prediction of the curve. Several properties of the optimal designs are derived. In general, D- and G-optimal designs are not equivalent. Optimal designs are determined numerically and compared with the uniform design.

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

  • Butler N.A. (2001) Properties of optimal polynomial designs under smoothing spline models. Journal of Statistical Planning and Inference 92: 259–266

    Article  MathSciNet  MATH  Google Scholar 

  • De Boor, C. (1978). A practical guide to splines. In Applied mathematical sciences (Vol. 27, XXIV). New York: Springer-Verlag.

  • Dette H., Melas V.B., Pepelyshev A.N. (2008) Optimal designs for free knot least squares splines. Statistica Sinica 18: 1047–1062

    MathSciNet  MATH  Google Scholar 

  • Dierckx, P. (1995). Curve and surface fitting with splines. In Monographs on numerical analysis. Oxford: Clarendon Press.

  • Efromovich S. (1999) Nonparametric curve estimation: Methods, theory and applications. Springer, New York

    MATH  Google Scholar 

  • Eubank, R. L. (1999). Nonparametric regression and spline smoothing (2nd ed.). In Statistics: Textbooks and monographs (Vol. 157). New York: Marcel Dekker.

  • Fan J., Gijbels I. (1996) Local polynomial modelling and its applications. Chapman and Hall, London

    MATH  Google Scholar 

  • Gantmacher F.R. (1998) The theory of matrices. Providence, Chelsea

    Google Scholar 

  • Green P., Silverman B. (1994) Nonparametric regression and generalized linear models. Chapman and Hall, London

    MATH  Google Scholar 

  • Heiligers B. (1998) E-optimal designs for polynomial spline regression. Journal of Statistical Planning and Inference 75: 159–172

    Article  MathSciNet  MATH  Google Scholar 

  • Heiligers B. (1999) Experimental design for polynomial spline regression. Tatra Mountains Mathematical Publications 17: 157–165

    MathSciNet  MATH  Google Scholar 

  • Jupp D.L.B. (1978) Approximation to data by splines with free knots. SIAM Journal on Numerical Analysis 15: 328–343

    Article  MathSciNet  MATH  Google Scholar 

  • Kaishev V.K. (1989) Optimal experimental designs for the B-spline regression. Computational Statistics & Data Analysis 8: 39–47

    Article  MathSciNet  MATH  Google Scholar 

  • Karlin S., Studden W.J. (1966) Tchebycheff systems: With applications in analysis and statistics. Wiley, New York

    MATH  Google Scholar 

  • Kiefer J. (1974) General equivalence theory for optimum designs (approximate theory). Annals of statistics 2: 849–879

    Article  MathSciNet  MATH  Google Scholar 

  • Kiefer J., Wolfowitz J. (1960) The equivalence of two extremum problems. Canadian Journal of Mathematics 12: 363–366

    Article  MathSciNet  MATH  Google Scholar 

  • Kuks J., Olman W. (1971) Minimaxnaja linejnaja ozenka coefficientov regressii II. Izvestiya Akademii Nauk Ehstonskoj SSR 21: 66–72

    MathSciNet  Google Scholar 

  • Mao W., Zhao L.H. (2003) Free-knot polynomial splines with confidence intervals. Journal of the Royal Statistical Society Series B 65(4): 901–919

    Article  MathSciNet  MATH  Google Scholar 

  • Murty V.N. (1971a) Optimal designs with a Tchebycheffian spline regression function. Annals of Mathematical Statistics 42: 643–649

    Article  MathSciNet  MATH  Google Scholar 

  • Murty V.N. (1971b) Optimal designs with a polynomial spline regression with a single multiple knot at the center. Annals of Mathematical Statistics 42: 952–960

    Article  MathSciNet  MATH  Google Scholar 

  • Park S.H. (1978) Experimental designs for fitting segmented polynomial regression models. Technometrics 20: 151–154

    Article  MATH  Google Scholar 

  • Pukelsheim F. (1993) Optimal design of experiments. Wiley, New York

    MATH  Google Scholar 

  • Reinsch C. (1967) Smoothing by spline functions. Numerische Mathematik 10: 177–183

    Article  MathSciNet  MATH  Google Scholar 

  • Schoenberg I.J. (1964) Spline functions and the problem of graduation. Proceedings of the National Academy of Sciences 52: 947–950

    Article  MathSciNet  MATH  Google Scholar 

  • Silverman B.W. (1985) Some aspects of the spline smoothing approach to non-parametric regression curve fitting. Journal of the Royal Statistical Society Series B 47: 1–52

    MATH  Google Scholar 

  • Silvey S.D. (1980) Optimal design. Chapman and Hall, London

    MATH  Google Scholar 

  • Studden, W. J. (1971). Optimal designs and spline regression. Optimizing methods in statistics. In Proceedings of the Symposium, Ohio State University, Columbus, Ohio (pp. 63–76). New York: Academic Press.

  • Studden W.J., VanArman D.J. (1969) Admissible designs for polynomial spline regression. Annals of Mathematical Statistics 40: 1557–1569

    Article  MathSciNet  MATH  Google Scholar 

  • Toutenburg H. (1982) Prior information in linear models. Wiley, New York

    MATH  Google Scholar 

  • Wahba, G. (1990). Spline models for observational data. In CBMS-NSF Regional Conference Series in Applied Mathematics (Vol. 59). Philadelphia: SIAM.

  • Whittaker E.T. (1923) On a new method of graduation. Proceedings of the Edinburgh Mathematical Society 41: 63–75

    Google Scholar 

  • Woods D.C. (2005) Designing experiments under random contamination with application to polynomial spline regression. Statistica Sinica 15: 619–635

    MathSciNet  MATH  Google Scholar 

  • Woods D., Lewis S. (2006) All-bias designs for polynomial spline regression models. Australian & New Zealand Journal of Statistics 48: 49–58

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Ruhr-Universität Bochum, 44780, Bochum, Germany

    Holger Dette

  2. Department of Mathematics, St. Petersburg State University, St. Petersburg, Russia

    Viatcheslav B. Melas & Andrey Pepelyshev

Authors
  1. Holger Dette
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Viatcheslav B. Melas
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andrey Pepelyshev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Holger Dette.

About this article

Cite this article

Dette, H., Melas, V.B. & Pepelyshev, A. Optimal design for smoothing splines. Ann Inst Stat Math 63, 981–1003 (2011). https://doi.org/10.1007/s10463-009-0265-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10463-009-0265-x

Keywords

Search

Navigation