Skip to main content
SpringerLink
Log in

Nonparametric models and their estimation

  • Articles
  • Published:
Allgemeines Statistisches Archiv Aims and scope Submit manuscript

Summary

Nonparametric models have become more and more popular over the last two decades. One reason for their popularity is software availability, which easily allows to fit smooth but otherwise unspecified functions to data. A benefit of the models is that the functional shape of a regression function is not prespecified in advance, but determined by the data. Clearly this allows for more insight which can be interpreted on a substance matter level.

This paper gives an overview of available fitting routines, commonly called smoothing procedures. Moreover, a number of extensions to classical scatterplot smoothing are discussed, with examples supporting the advantages of the routines.

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

  • Akaike, H. (1970). Statistical predictor identification. Annals of the Institute of Statistical Mathematics 22 203–217.

    Article  MATH  MathSciNet  Google Scholar 

  • Akritas, M., Politis, D. (2003). Recent Advances and Trends in Nonparametric Statistics. North Holland, Amsterdam.

    Google Scholar 

  • Bowman, A. W., Azzalini, A. (1997). Applied Smoothing Techniques for Data Analysis: the Kernel Approach with S-Plus Illustrations. Oxford University Press, Oxford.

    Google Scholar 

  • Chiarella, C., Flaschel, P. (2000). The Dynamics of Keynesian Monetary Growth: Macro Foundations. Cambridge University Press, Cambridge.

    Google Scholar 

  • Cox, D. R. (1972). Regression models and life tables (with discussion). Journal of the Royal Statistical Association, Series B 34 187–220.

    MATH  Google Scholar 

  • Crainiceanu, C., Ruppert, D., Claeskens, G., Wand, M. (2005). Exact likelihood ratio test for penalized splines. Biometrika 92 91–103.

    Article  MATH  MathSciNet  Google Scholar 

  • Dalgaard, P. (2002). Introductory Statistics with R. Springer, New York.

    MATH  Google Scholar 

  • de Boor (1978). A Practical Guide to Splines. Springer, Berlin.

    MATH  Google Scholar 

  • Efron, B. (2001). Selection criteria for scatterplot smoothers. The Annals of Statistics 29 470–504.

    Article  MATH  MathSciNet  Google Scholar 

  • Eilers, P., Marx, B. (1996). Flexible smoothing with B-splines and penalties. Statistical Science 11 89–121.

    Article  MATH  MathSciNet  Google Scholar 

  • Eubank, R. L. (1988) Spline Smoothing and Nonparametric Regression. Dekker, New York.

    MATH  Google Scholar 

  • Fahrmeir, L., Tutz, G. (2001). Multivariate Statistical Modelling Based on Generalized Linear Models. 2nd ed., Springer, New York.

    MATH  Google Scholar 

  • Fan, J., Gijbels, I. (1996). Local Polynomial Modelling and its Applications. Chapman and Hall, London.

    MATH  Google Scholar 

  • Fan, J., Yao, O. (2003). Nonlinear Time Series: Nonparametric and Parametric Methods. Springer, New York.

    Google Scholar 

  • Flaschel, P., Kauermann, G., Semmler, W. (2005). Testing wage and price Phillips curves for the United States. Metroeconomica (to appear).

  • Grambsch, P. M., Therneau, T. M. (2000). Modelling Survival Data: Extending the Cox Model. Springer, New York.

    Google Scholar 

  • Green, D. J., Silverman, B. W. (1994). Nonparametric Regression and Generalized Linear Models. Chapman and Hall, London.

    MATH  Google Scholar 

  • Gu, C., Wahba, G. (1991). Smoothing spline ANOVA with component-wise Bayesian confidence intervals. Journal of Computational and Graphical Statistics 2 97–117.

    Article  MathSciNet  Google Scholar 

  • Härdle, W., Hall, W., Marron, J. S. (1988). How far are automatically chosen regression smoothing parameter selectors from their optimum? Journal of the American Statistical Association 83 86–101.

    Article  MATH  MathSciNet  Google Scholar 

  • Härdle, W., Hall, W., Marron, J. S. (1992). Regression smoothing parameters that are not far from their optimum. Journal of the American Statistical Association 87 227–233.

    Article  MATH  MathSciNet  Google Scholar 

  • Härdle, W., Mammen, E. (1993). Comparing nonparametric versus parametric regression fits. The Annals of Statistics 21 1926–1947.

    MATH  MathSciNet  Google Scholar 

  • Härdle, W., Lütkepohl, H., Chen, R. (1997). A review of nonparametric time series analysis. International Statistical Review 65 49–72.

    MATH  Google Scholar 

  • Härdle, W., Hlavka, Z., Klinke, S. (2000). XploRe, Application Guide. Springer, Berlin.

    MATH  Google Scholar 

  • Härdle, W., Müller, M., Sperlich, S., Werwatz, A. (2004). Nonparametric and Semiparametric Models. Springer, Berlin.

    MATH  Google Scholar 

  • Hastie, T. (1996). Pseudosplines. Journal of the Royal Statistical Society Series B 58 379–396.

    MATH  MathSciNet  Google Scholar 

  • Hastie, T., Tibshirani, R. (1990). Generalized Additive Models. Chapman and Hall, London.

    MATH  Google Scholar 

  • Hastie, T., Tibshirani, R. (1993). Varying-coefficient models. Journal of the Royal Statistical Society, Series B 55 757–796.

    MATH  MathSciNet  Google Scholar 

  • Hurvich, C. M., Simonoff, J. S., Tsai, C. (1998). Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion. Journal of the Royal Statistical Society, Series B 60 271–293.

    Article  MATH  MathSciNet  Google Scholar 

  • Kauermann, G. (2000). Modelling longitudinal data with ordinal response by varying coefficients. Biometrics 56 692–698.

    Article  MATH  Google Scholar 

  • Kauermann, G. (2004). A note on smoothing parameter selection for penalized spline smoothing. Journal of Statistical Planing and Inference 127 53–69.

    Article  MathSciNet  Google Scholar 

  • Kauermann, G. (2005). Penalized spline fitting in multivariable survival models with varying coefficients. Computational Statistics and Data Analysis 49 169–186.

    Article  MathSciNet  Google Scholar 

  • Kauermann, G., Tutz, G. (2000). Local likelihood estimation in varying-coefficient models including additive bias correction. Journal of Nonparametric Statistics 12 343–371.

    MATH  MathSciNet  Google Scholar 

  • Kauermann, G., Tutz, G. (2001). Testing generalized linear and semiparametric models against smooth alternatives. Journal of the Royal Statistical Society, Series B 63 147–166.

    Article  MATH  MathSciNet  Google Scholar 

  • Kauermann, G., Tutz, G., Brüderl, J. (2005). The survival of newly founded firms: A case study into varying-coefficient models. Journal of the Royal Statistical Society, Series A 168 145–158.

    Google Scholar 

  • Loader, C. (1999). Local Regression and Likelihood. Springer, Berlin.

    MATH  Google Scholar 

  • Marron, J. S., Chaudhuri, P. (1999). SiZer for exploration of structures in curves. Journal of the American Statistical Association 94 807–823.

    Article  MATH  MathSciNet  Google Scholar 

  • McCullagh, P., Nelder, J. A. (1989). Generalized Linear Models. 2nd ed., Chapman and Hall, New York.

    MATH  Google Scholar 

  • Nadaraya, E. A. (1964). On estimating regression. Theory of Probability and Application 9 141–142.

    Article  Google Scholar 

  • Nychka, D. (2000). Spatial process estimates as smoothers. In Smoothing and Regression. Approaches, Computation and Application (Schimek, ed.), Wiley, New York.

    Google Scholar 

  • O'Sullivan, F. (1986). A statistical perspective on ill-posed inverse problems. Statistical Science 1 502–518.

    MATH  MathSciNet  Google Scholar 

  • Opsomer, J. D., Wang, Y., Yang, Y. (2001). Nonparametric regression with correlated errors. Statistical Science 16 134–153.

    Article  MATH  MathSciNet  Google Scholar 

  • Pagan, R., Ullah, A. (1999). Nonparametric Econometrics. Cambridge University Press, Cambridge.

    Google Scholar 

  • Phillips, A. W. (1958). The relation between unemployment and the rate of change of money wage rates in the United Kingdom, 1861–1957. Economica 25 283–299.

    Article  Google Scholar 

  • Reinsch, C. H. (1967). Smoothing by spline functions. Numerical Mathematics 10 177–183.

    Article  MATH  MathSciNet  Google Scholar 

  • Rice, J. A. (1984). Bandwidth choice for nonparametric regression. Annals of Statistics 12 1215–1230.

    MATH  MathSciNet  Google Scholar 

  • Ripley, B. D., Venables, W. N. (2002). Modern Applied Statistics with S. 4th ed., Springer, New York.

    Google Scholar 

  • Ruppert, D. (2004). Statistics and Finance. Springer, New York.

    MATH  Google Scholar 

  • Ruppert, R., Wand, M. P., Carroll, R. J. (2003). Semiparametric Regression. Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  • Simonoff, J. S. (1996). Smoothing Methods in Statistics. Springer, New York.

    MATH  Google Scholar 

  • Stone, M. (1974). Cross-validatory choice and assessment of statistical predictions. Journal of the Royal Statistical Society, Series B 36 111–147.

    MATH  Google Scholar 

  • Wand, M. P. (2003). Smoothing and mixed models. Computational Statistics 18 223–249.

    MATH  Google Scholar 

  • Wahba, G. (1990). Regularization and cross validation methods for nonlinear implicit, ill-posed inverse problems. In Geophysical Data Inversion Methods and Applications (A. Vogel, C. Ofoegbu, R. Gorenflo and B. Ursin, eds.), 3–13. Wieweg, Wiesbaden-Braunschweig.

    Google Scholar 

  • Watson, G. S. (1964). Smooth regression analysis. Sankhyā, Series A 26 359–372.

    MATH  Google Scholar 

  • Wood, S. N. (2003). Thin plate regression splines. Journal of the Royal Statistical Society, Series B 65 95–114.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Business Administration and Economics, University Bielefeld, P. O. Box 100131, 33502, Bielefeld

    Göran Kauermann

Authors
  1. Göran Kauermann
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kauermann, G. Nonparametric models and their estimation. Allgemeines Statistisches Arch 90, 137–152 (2006). https://doi.org/10.1007/s10182-006-0226-0

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10182-006-0226-0

Keywords

Search

Navigation