Skip to main content
Log in

No effect and lack-of-fit permutation tests for functional regression

  • Original Paper
  • Published:
Computational Statistics Aims and scope Submit manuscript

Abstract

This paper proposes statistical procedures to check if a real-valued covariate X has an effect on a functional response Y(t). A non-parametric kernel regression is considered to estimate the influence of X on Y(t) and two test statistics based on residual sums of squares and smoothing residuals are proposed. Their acceptance levels are determined by means of permutations. The lack-of-fit test for a class of parametric models is then discussed as a consequence of the no effect procedure. Monte Carlo simulations provide an insight into the level and the power of the no effect tests. A study of atmospheric radiation illustrates the behavior of the proposed methods in practice.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Aguilera AM, Ocaa FA, Valderrama MJ (1999) Forecasting with unequally spaced data by a functional principal component approach. Test 8:233–253

    Article  MATH  Google Scholar 

  • Antoch J, Hlubinka D, Prchal L (2006) Statistical methods for analysis of meteorological measurements (submitted)

  • Benko M, Härdle W, Kneip A (2006) Common functional principal components. SFB649 economic risk discussion paper 2006–2010. Humboldt University, Berlin

  • Besse P, Cardot H, Ferraty F (1997) Simultaneous non-parametric regressions of unbalanced longitudinal data. Comput Stat Data Anal 24:255–270

    Article  MATH  Google Scholar 

  • Bowman AW, Azzalini A (1997) Applied smoothing techniques for data analysis. Clarendon Press, UK

    MATH  Google Scholar 

  • Cardot H (2006) Conditional functional principal components analysis. Scand J Stat (to appear)

  • Cardot H, Sarda P (2005) Estimation in generalized linear models for functional data via penalized likelihood. J Multivar Anal 92:24–41

    Article  MATH  Google Scholar 

  • Cardot H, Ferraty F, Sarda P (1999) Functional linear model. Stat Prob Lett 45:11–22

    Article  MATH  Google Scholar 

  • Cardot H, Ferraty F, Mas A, Sarda P (2003) Testing hypotheses in the functional linear model. Scand J Stat 30:241–255

    Article  MATH  Google Scholar 

  • Cardot H, Goia A, Sarda P (2004) Testing for no effect in functional linear regression models, some computational approaches. Comm Stat Simulat Comput 33:179–199

    Article  MATH  Google Scholar 

  • Chiou JM, Müller HG, Wang JL (2004) Functional response models. Stat Sin 14:659–677

    Google Scholar 

  • Cuevas A, Fraiman R (2004) On the bootstrap methodology for functional data. In: Antoch J (ed) Proceedings in computational statistics, COMPSTAT 2004. Physica-Verlag, Heidelberg, pp 127–135

  • Cuevas A, Febrero M, Fraiman R (2004) An anova test for functional data. Comput Stat Data Anal 47:111–122

    Article  Google Scholar 

  • Deville JC (1974) Méthodes statistiques et numériques de l’analyse harmonique. Ann Insee 15:3–104

    Google Scholar 

  • Escabias M, Aguilera AM, Valderrama MJ (2004) Principal component estimation of functional logistic regression: discussion of two different approaches. J Nonparametr Stat 16:365–384

    Article  MATH  Google Scholar 

  • Ferraty F, Vieu P (2002) The functional non-parametric model and application to spectrometric data. Comput Stat 17:545–564

    Article  MATH  Google Scholar 

  • Ferraty F, Vieu P (2006) Non-parametric functional data analysis: theory and practice. Springer, Heidelberg

    Google Scholar 

  • Good P (2000) Permutation tests. A practital guide to resampling methods for testing hypotheses, 2nd edn. Springer, Heidelberg

    Google Scholar 

  • Hart JD (1997) Non-parametric smoothing and lack-of-fit tests. Springer, Heidelberg

    Google Scholar 

  • Härdle W, Marron JS (1985) Optimal bandwidth selection in non-parametric regression function estimation. Ann Stat 13:1465–1481

    MATH  Google Scholar 

  • Hlubinka D, Prchal L (2006) Changes in atmospheric radiation from the statistical point of view. Comput Stat Data Anal (in press)

  • James GM (2002) Generalized linear models with functional predictors. J R Stat Soc Ser B Stat Methodol 64:411–432

    Article  MATH  Google Scholar 

  • Lehmann EL (1959) Testing statistical hypotheses. Wiley, London

    MATH  Google Scholar 

  • Marx BD, Eilers PH (1999) Generalized linear regression on sampled signals and curves: a p-spline approach. Technometrics 41:1–13

    Article  Google Scholar 

  • Müller HG, Stadtmüller U (2005) Generalized functional linear models. Ann Stat 33:774–805

    Article  MATH  Google Scholar 

  • Ramsay JO, Dalzell CJ (1991) Some tools for functional data analysis. J R Stat Soc Ser B Stat Methodol 52:539–572

    Google Scholar 

  • Ramsay JO, Silverman BW (2002) Applied functional data analysis: methods and case studies. Springer, Heidelberg

    MATH  Google Scholar 

  • Ramsay JO, Silverman BW (2005) Functional data analysis, 2nd edn. Springer, Heidelberg

    Google Scholar 

  • Raz J (1990) Testing for no effect when estimating a smooth function by non-parametric regression: a randomization approach. J Am Stat Assoc 85:132–138

    Article  Google Scholar 

  • Staniswalis JG, Lee JJ (1998) Non-parametric regression analysis of longitudinal data. J Am Stat Assoc 93:1403–1418

    Article  MATH  Google Scholar 

  • Yao F, Müller HG, Wang JL (2005) Functional data analysis for sparse longitudinal data. J Am Stat Assoc 100:577–590

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hervé Cardot.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cardot, H., Prchal, L. & Sarda, P. No effect and lack-of-fit permutation tests for functional regression. Computational Statistics 22, 371–390 (2007). https://doi.org/10.1007/s00180-007-0046-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00180-007-0046-z

Keywords

Navigation