Skip to main content
SpringerLink
Log in

Improved Model Selection Method for a Regression Function with Dependent Noise

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

Abstract

This paper is devoted to nonparametric estimation, through the \(\mathcal{L}_2\)-risk, of a regression function based on observations with spherically symmetric errors, which are dependent random variables (except in the normal case). We apply a model selection approach using improved estimates. In a nonasymptotic setting, an upper bound for the risk is obtained (oracle inequality). Moreover asymptotic properties are given, such as upper and lower bounds for the risk, which provide optimal rate of convergence for penalized estimators.

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

    Article  MATH  Google Scholar 

  2. Beran J. (1992). Statistical methods for data with long-range depedence. Statistical Science, 7: 404–427

    Google Scholar 

  3. Berger J.O. (1975). Minimax estimation of location vectors for a wide class of densities. The Annals of Statistics, 3(6): 1318–1328

    MATH  Google Scholar 

  4. Blanchard D., Fourdrinier D. (1999). Non trivial solutions of non-linear partial differential inequations and order cut-off. Rendiconti di Matematica, Serie VII, 19: 137–154

    MATH  Google Scholar 

  5. Brown L.D. (1966). On the admissibility of invariant estimators of one or more location parameters. The Annals of Mathematical Statistics, 37: 1087–1136

    Google Scholar 

  6. Csörgó S., Mielniczuk J. (1995). Nonparametric regression under longe range dependent normal errors. The Annals of Statistics, 23(3): 1000–1014

    Google Scholar 

  7. Dahlhaus R. (1995). Efficient location and regression estimation for longrange dependent regression models. The Annals of Statistics, 23(3): 1029–1047

    MATH  Google Scholar 

  8. Donoho D.L., Johnstone I.M., Kerkyacharian G., Picard D. (1995). Wavelet shrinkage: asymptotia?. Journal of Royal Statistical Society, Services B, 57: 301–369

    MATH  Google Scholar 

  9. Efroimovich S.Yu. (1999). Nonparametric curve estimation. Methods, theory and applications. Springer, Berlin Haidelberg New York

    Google Scholar 

  10. Fang K.-T., Kotz S., Ng K.-W. (1989). Symmetric Multivariate and Related Distributions. Chapman and Hall, New york

    Google Scholar 

  11. Fourdrinier D., Strawderman W.E. (1996). A paradox concerning shrinkage estimators: should a known scale parameter be replaced by an estimated value in the shrinkage factor?. Journal of Multivariate Analysis, 59(2): 109–140

    Article  MATH  Google Scholar 

  12. Fourdrinier D., Wells M.T. (1994). Comparaison de procédures de sélection d’un modèle de régression: une approche décisionnelle. Comptes Rendus de l’Académie des Sciences Paris, v. 319, série I, p. 865–870

  13. Gunst R.F., Mason R.L. (1980). Regression analysis and its applications: a data oriented approach. New York, Marcel Dekker

    MATH  Google Scholar 

  14. James W., Stein C. (1961). Estimation with quadratic loss. In Proceedings of the Fourth Berkeley Symposium Mathematics, Statistics and Probability, University of California Press, Berkeley, 1: 361–380

  15. Hall P., Hart J. (1990). Nonparametric regression with long range dependence. Stochastics Processes and their Applications 36: 339–351

    Article  MATH  Google Scholar 

  16. Kariya T., Sinha B.K. (1993). The robustness of statistical tests. Academic, New York

    Google Scholar 

  17. Kotz S. (1975). Multivariate distributions at a cross-road. In Patil G.J., Kotz S., Ord J.K. Statistical Distributions in Scientific Work. (Eds.) 1: 245–270

  18. Nemirovski A. (2000). Topics in non-parametric statistics. Lectures on probability theory and statistics, Saint-Flour 1998. In Lecture Notes in Mathematics, 1738. Berlin Heidelberg New York: Springer, pp 85–277

  19. Stein C.M. (1956). Inadmissibility of usual estimator of the mean of a multivariate distribution. In Proceedings of the 3rd Berkely Symposium in Mathematical Statistics and Probability, 1 197–206

  20. Stein C.M. (1981). Estimation of the mean of a multivariate normal distribution. The Annals of Statistics, 9: 1135–1151

    MATH  Google Scholar 

  21. Timan A.F. (1963). Theory of approximation of functions of a real variable. Pergamon, Oxford

    MATH  Google Scholar 

  22. Tsybakov A. (1998). Pointwise and sup-norm sharp adaptive estimation of functions on the Sobolev classes. The Annals of Statistics, 26(6): 221–245

    Google Scholar 

  23. Ziemer W.P. (1989). Weakly differentiable functions—sobolev spaces and functions of bounded variation. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. UMR CNRS 6085, Laboratoire de Mathématiques Raphaël Salem Université de Rouen, Avenue de l’Université BP. 12, 76801, Saint-Étienne-du-Rouvray, France

    D. Fourdrinier & S. Pergamenshchikov

Authors
  1. D. Fourdrinier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Pergamenshchikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Fourdrinier.

About this article

Cite this article

Fourdrinier, D., Pergamenshchikov, S. Improved Model Selection Method for a Regression Function with Dependent Noise. AISM 59, 435–464 (2007). https://doi.org/10.1007/s10463-006-0063-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10463-006-0063-7

Keywords

Search

Navigation