Skip to main content
SpringerLink
Log in

Testing linearity in semi-parametric functional data analysis

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

Abstract

This paper investigates a semi-parametric model for functional data, based on partial linear ideas. A methodology is developped for testing the linear component of such a functional partial linear model. The behavior of the test is studied through some finite simulated samples before being applied to some chemometrical curves dataset. One interesting feature of the methodology is that it works with not necessarily independent data.

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

  • Aguilera A, Escabias M, Valderrama M (2008) Forecasting binary longitudinal data by a functional PC-ARIMA model. Comput Stat Data Anal 52: 3187–3197

    Article  MathSciNet  MATH  Google Scholar 

  • Ait Saidi A, Ferraty F, Kassa R, Vieu P (2008) Cross-validated estimations in the single functional index model. Statistics 42(6): 475–494

    Article  MathSciNet  MATH  Google Scholar 

  • Aneiros-Pérez G, Vieu P (2006) Semi-functional partial linear regression. Stat Probab Lett 11: 1102–1110

    Article  Google Scholar 

  • Aneiros-Pérez G, Vieu P (2008) Nonparametric time series prediction: a semi-functional partial linear modeling. J Multivar Anal 99: 834–857

    Article  MATH  Google Scholar 

  • Aneiros-Pérez G, Vieu P (2011) Automatic estimation procedure in partial linear model with functional data. Stat Papers 52: 751–771

    Article  MATH  Google Scholar 

  • Aneiros-Pérez G, Cardot H, Estévez-Pérez G, Vieu P (2004a) Maximum ozone concentration forecasting by functional nonparametric approaches. Environmetrics 15: 675–685

    Article  Google Scholar 

  • Aneiros-Pérez G, González-Manteiga W, Vieu P (2004b) Estimation and testing in a partial linear regression model under long-memory dependence. Bernoulli 10: 49–78

    Article  MathSciNet  MATH  Google Scholar 

  • Beran J, Ghosh S (1998) Root-n-consistent estimation in partial linear models with long-memory errors. Scand J Stat 25: 345–357

    Article  MathSciNet  MATH  Google Scholar 

  • Bickel P, Ritov Y, Stoker T (1993) Efficient and adaptive estimation for semiparametric models. John Hopkins Series in the Math. Sci., Johns Hopkins University Press, Baltimore

  • Cardot H, Sarda P (2011) Functional linear regression. In: Ferraty F, Romain Y (eds) The oxford handbook of functional data analysis. Oxford University Press, New York, pp 21–46

    Google Scholar 

  • Chiou JM, Müller HG (2007) Diagnostics for functional regression via residual processes. Comput Stat Data Anal 51(10): 4849–4863

    Article  MATH  Google Scholar 

  • Davidian M, Lin X, Wang JL (2004) Introduction to the emerging issues in longitudinal and functional data analysis (with discussion). Stat Sin 14(3): 613–629

    MathSciNet  Google Scholar 

  • Domowitz J (1982) The linear model with stochastic regressors and heteroscedastic dependent errors. Discussion paper No 543, Center for Mathematical studies in Economic and Management Science, Northwestern University, Evanston, Illinois

  • Ferraty F (2010) Special issue on statistical methods and problems in infinite dimensional spaces. J Multivar Anal 101(2): 305–490

    Article  MathSciNet  Google Scholar 

  • Ferraty F, Vieu P (2006) Nonparametric functional data analysis: theory and practice. Springer Series in Statistics, New York

    MATH  Google Scholar 

  • Ferraty F, Vieu P (2009) Additive prediction and boosting for functional data. Comput Stat Data Anal 53(10): 1400–1413

    Article  MathSciNet  MATH  Google Scholar 

  • Ferraty F, Romain Y (2011) The oxford handbook of functional data analysis. Oxford University Press, New York

    Google Scholar 

  • Ferraty F, Vieu P (2011a) A unifying classification for functional regression modeling. In: Ferraty F, Romain Y (eds) The oxford handbook of functional data analysis. Oxford University Press, New York, pp 3–20

    Google Scholar 

  • Ferraty F, Vieu P (2011b) Kernel regression estimation for functional data. In: Ferraty F, Romain Y (eds) The oxford handbook of functional data analysis. Oxford University Press, New York, pp 72–129

    Google Scholar 

  • Ferraty F, Mas A, Vieu P (2007) Nonparametric regression of functional data: inference and practical aspects. Aust N Z J Stat 49(3): 267–286

    Article  MathSciNet  MATH  Google Scholar 

  • González-Manteiga W, Aneiros-Pérez G (2003) Testing in partial linear regression models with dependent errors. J Nonparametr Stat 15: 93–111

    Article  MathSciNet  MATH  Google Scholar 

  • González-Manteiga W, Vieu P (2007) Introduction to the Special Issue on Statistics for Functional Data. Comput Stat Data Anal 51(10): 4788–4792

    Article  Google Scholar 

  • Gozalo P, Linton O (2001) Testing additivity in generalized nonparametric regression models with estimated parameters. J Econom 104: 148

    Article  MathSciNet  Google Scholar 

  • Härdle W, Liang H, Gao J (2000) Partially linear models. Springer series in statistics. Springer, New York

  • Härdle W, Müller M, Sperlich S, Werwatz A (2004) Nonparametric and semiparametric models. Contributions to statistics. Physica Verlag, Heidelberg

  • Härdle W, Mori Y, Vieu P (2007) Statistical methods for biostatistics and related fields. Contributions to statistics. Springer, Heidelberg

  • Hastie T, Tibshirani R (1990) Generalized additive models. Chapman and Hall, London

    MATH  Google Scholar 

  • Hlubinka D, Prchal L (2007) Changes in atmospheric radiation from the statistical point of view. Comput Stat Data Anal 51(10): 4926–4941

    Article  MathSciNet  MATH  Google Scholar 

  • Horowitz J (1998) Semiparametric methods in econometrics. Lecture Notes in Statistics, 131, Springer Verlag, New York

  • Hyndman R, Shahid Ullah M (2007) Robust forecasting of mortality and fertility rates: a functional data approach. Comput Stat Data Anal 51(10): 4942–4956

    Article  MathSciNet  MATH  Google Scholar 

  • James G (2011) Sparseness and functional data analysis. In: Ferraty F, Romain Y (eds) The oxford handbook of functional data analysis. Oxford University Press, New York, pp 298–323

    Google Scholar 

  • Judge GG, Griffiths WE, Carter R, Lütkepohl H, Lee TC (1985) The theory and practice of econometrics. Wiley, New York

    Google Scholar 

  • Kauermann G, Carroll RJ (2001) A note on the efficiency of sandwich covariance matrix estimation. J Am Stat Assoc 96: 1387–1396

    Article  MathSciNet  MATH  Google Scholar 

  • Ling S, Li WK (1997) On fractionally integrated autoregressive moving-average time series models with conditional heteroscedasticity. J Am Stat Assoc 92: 1184–1194

    Article  MathSciNet  MATH  Google Scholar 

  • López-Pintado S, Romo J (2007) Depth-based inference for functional data. Comput Stat Data Anal 51(10): 4957–4968

    Article  MATH  Google Scholar 

  • Manté C, Yao AF, Degiovanni C (2007) Depth-based inference for functional data. Comput Stat Data Anal 51(10): 4969–4983

    Article  MATH  Google Scholar 

  • Nérini D, Ghattas B (2007) Classifying densities using functional regression trees: Application in oceanology. Comput Stat Data Anal 51(10): 4984–4993

    Article  MATH  Google Scholar 

  • Pelegrina L, Sarda P, Vieu P (1996) On multidimensional nonparametric regression. In: Prat A (ed) Proceedings in Computational Statistics, COMPSTAT 1996. Physica Heidelberg, pp 149–160

  • Rachdi M, Vieu P (2007) Nonparametric regression for functional data: automatic smoothing parameter selection. J Stat Plan Inference 137(9): 2784–2801

    Article  MathSciNet  MATH  Google Scholar 

  • Ramsay J, Silverman B (2005) Functional data analysis, 2nd en. Springer series in statistics. Springer, New York

  • Robinson P (1988) Root-n-consistent semi-parametric regression. Econometrica 56: 931–954

    Article  MathSciNet  MATH  Google Scholar 

  • Ruppert D, Wand M, Carroll R (2003) Semiparametric regression. Cambridge series in statistical and probabilistic mathematics, vol 12. Cambridge University Press, Cambridge

  • Schimek M (2000) Smoothing and regression. Approaches, computation and applications. Wiley series in probability and statistics. Wiley, New York

  • Shin H (2009) Partial functional linear regression. J Stat Plan Inference 139: 3405–3418

    Article  MATH  Google Scholar 

  • Speckman P (1988) Kernel smoothing in partial linear models. J R Stat Soc Ser B 50: 413–436

    MathSciNet  MATH  Google Scholar 

  • Sperlich S, Härdle W, Gökhan A (2003) The art of semiparametrics. Contributions to statistics, Physica Verlag, Springer, Heidelberg

  • Stone C (1985) Additive regression and other nonparametric models. Ann Stat 13: 689–705

    Article  MATH  Google Scholar 

  • Valderrama M (2007) Introduction to the special issue on modelling functional data in practice. Comput Stat 22(3): 331–334

    Article  MathSciNet  Google Scholar 

  • Zhou J, Zhu Z, Fung WK (2008) Robust testing with generalized partial linear models for longitudinal data. J Stat Plan Inference 138: 1871–1883

    Article  MathSciNet  MATH  Google Scholar 

  • Zhu LX, Ng KW (2003) Checking the adequacy of a partial linear model. Stat Sin 13: 763–781

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universidad de A Coruña, A Coruña, Spain

    Germán Aneiros-Pérez

  2. Université Paul Sabatier, Toulouse, France

    Philippe Vieu

Authors
  1. Germán Aneiros-Pérez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Philippe Vieu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Germán Aneiros-Pérez.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aneiros-Pérez, G., Vieu, P. Testing linearity in semi-parametric functional data analysis. Comput Stat 28, 413–434 (2013). https://doi.org/10.1007/s00180-012-0308-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00180-012-0308-2

Keywords

Search

Navigation