Skip to main content
SpringerLink
Log in

On the optimal choice of the number of empirical Fourier coefficients for comparison of regression curves

  • Regular Article
  • Published:
Statistical Papers Aims and scope Submit manuscript

Abstract

The paper is devoted to the elaboration of an efficient approach for comparison of two regression curves based on the empirical Fourier coefficients of regression functions. For the problem of testing for the equality of the two unknown functions in the case of homoscedastic error structure and observation at equidistant points, we derive a new procedure with adaptive choice of the number of the coefficients used in the hypotheses testing. Our approach is based on approximation of the most powerful test using the full knowledge of the regression functions. The results are justified by theoretical arguments and the superiority of the new procedure is also confirmed by a simulation study.

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

Fig. 1
Fig. 2

Similar content being viewed by others

References

  • Azzalini A, Bowman A (1993) On the use of nonparametric regression for checking linear relationships. J R Stat Soc Ser B 55:549–557

    MATH  MathSciNet  Google Scholar 

  • Cox D, Koh G, Wahba G, Yandell BS (1988) Testing the (parametric) null model hypothesis in (semiparametric) partial and generalized spline models. Ann Stat 16:113–119

    Article  MATH  MathSciNet  Google Scholar 

  • Delgado MA (1993) Testing the equality of nonparametric regression curves. Stat Probab Lett 17:199–204

    Article  MATH  MathSciNet  Google Scholar 

  • Dette H, Neumeyer N (2001) Nonparametric analysis of covariance. Ann Stat 29:1361–1400

    Article  MATH  MathSciNet  Google Scholar 

  • Eubank RL, Hart JD (1992) Testing goodness-of-fit in regression via order selection criteria. Ann Stat 20:1412–1425

    Article  MATH  MathSciNet  Google Scholar 

  • Eubank RL, Spiegelman CH (1990) Testing the goodness of fit of a linear model via nonparametric regression techniques. J Am Stat Assoc 85:387–392

    Article  MATH  MathSciNet  Google Scholar 

  • Fan J, Huang L-S (2001) Goodness-of-fit tests for parametric regression models. J Am Stat Assoc 96:640–652

    Article  MATH  MathSciNet  Google Scholar 

  • Fan J (1996) Test of significance bsed on Wavelet thresholding and Neyman’s truncation. J Am Stat Assoc 93:1007–1021

    Article  Google Scholar 

  • Ferraty F, Vieu P (2006) Nonparametric functional data analysis. Springer, New York

    MATH  Google Scholar 

  • Gradshteyn IS, Ryzhik IM (2007) Table of integrals, series, and products, seventh edn. Academic Press, Amsterdam

    MATH  Google Scholar 

  • Hall P, Hart JD (1990) Bootstrap test for difference between means in nonparametric regression. J Am Stat Assoc 85:1039–1049

    Article  MATH  MathSciNet  Google Scholar 

  • Hardle W, Mammen E (1993) Comparing nonparametric versus parametric regression fits. Ann Stat 21(4):1926–1947

    Article  MathSciNet  Google Scholar 

  • Härdle W, Marron JS (1990) Semiparametric comparison of regression curves. Ann Stat 18:83–89

    Article  Google Scholar 

  • King EC, Hart JD, Wehrly TE (1991) Testing the equality of two regression curves using linear smoothers. Stat Probab Lett 12:239–247

    Article  MathSciNet  Google Scholar 

  • Kulasekera KB, Wang J (1997) Smoothing parameter selection for power optimality in testing of regression curves. J Am Stat Assoc 92:500–511

    Article  MATH  MathSciNet  Google Scholar 

  • Liang H (2004) Comparison of curves based on a Cramer-von Mises statistic. Comput Stat Data Anal 45(4):805–812

    Article  MATH  Google Scholar 

  • Ledwina T, Wylupek G (2012) Two-sample test against one-sided alternatives. Scand J Stat 39(2):358–381

    Article  MATH  MathSciNet  Google Scholar 

  • Mohdeb Z, Mezhoud KA, Boudaa D (2010) Testing the equality of nonparametric regression curves based on Fourier coefficients. J Afr Stat 5(4):219–227

    MATH  MathSciNet  Google Scholar 

  • Muirhead R (2005) Aspects of multivariate statistical theory, 2nd edn. Wiley, New York

    Google Scholar 

  • Munk A, Dette H (1998) Nonparametric comparison of several regression functions: exact and asymptotic theory. Ann Stat 26:2339–2368

    Article  MATH  MathSciNet  Google Scholar 

  • Neumeyer N, Dette H (2003) Nonparametric comparison of regression curves: an empirical process approach. Ann Stat 31(3):880–920

    Article  MATH  MathSciNet  Google Scholar 

  • Ramsay JO, Silverman BW (1997) Functional data analysis. Springer, New York

    Book  MATH  Google Scholar 

  • Rice JA (1984) Bandwidth choice for nonparametric regression. Ann Stat 12:1215–1230

    Article  MATH  MathSciNet  Google Scholar 

  • Vilar-Fernandez JM, Vilar-Fernandez JA, Gonzalez-Manteiga W (2007) Bootstrap tests for nonparametric comparison of regression curves with dependent errors. TEST 16:123–144

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work was initiated during the visit of Viatcheslav Melas at the Department of Management and Engineering, University of Padova, as a Visiting Scientist in July 2012. The work of Viatcheslav Melas, Petr Shpilev and Andrey Pepelyshev was supported in part by Russian Foundation of Basic Research, project 12-01-00747. Authors wish to thank Fortunato Pesarin for stimulating discussion on topics related to this paper. The paper has to be considered as a joint work where all authors have contributed equally likely.

Author information

Authors and Affiliations

  1. Faculty of Mathematics & Mechanics, St.Petersburg State University, 28, University avenue, Petrodvoretz, 198504, St.Petersburg, Russia

    Viatcheslav B. Melas, Andrey Pepelyshev & Petr Shpilev

  2. Department of Management and Engineering, University of Padova, Stradella S. Nicola, 3, 36100, Vicenza, Italy

    Luigi Salmaso & Livio Corain

  3. Department of Land, Environment, Agriculture and Forestry, University of Padova, Viale dell’University, 16, Legnaro, 35020, Padua, Italy

    Rosa Arboretti

Authors
  1. Viatcheslav B. Melas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrey Pepelyshev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Petr Shpilev
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Luigi Salmaso
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Livio Corain
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Rosa Arboretti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luigi Salmaso.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Melas, V.B., Pepelyshev, A., Shpilev, P. et al. On the optimal choice of the number of empirical Fourier coefficients for comparison of regression curves. Stat Papers 56, 981–997 (2015). https://doi.org/10.1007/s00362-014-0619-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00362-014-0619-1

Keywords

Mathematics Subject Classification

Search

Navigation