Download PDF

Feasible estimation in generalized structured models

Statistics and Computing volume 20, pages367–379(2010)Cite this article

Abstract

This article introduces a feasible estimation method for a large class of semi and nonparametric models. We present the family of generalized structured models which we wish to estimate. After highlighting the main idea of the theoretical smooth backfitting estimators, we introduce a general estimation procedure. We consider modifications and practical issues, and discuss inference, cross validation, and asymptotic theory applying the theoretical framework of Mammen and Nielsen (Biometrika 90: 551–566, 2003). An extensive simulation study shows excellent performance of our method. Furthermore, real data applications from environmetrics and biometrics demonstrate its usefulness.

References

  1. Binder, H., Tutz, G.: A comparison of methods for the fitting of generalized additive models. Stat. Comput. 18, 87–99 (2008)

    Article  MathSciNet  Google Scholar 

  2. Cadarso-Suárez, C., Roca-Pardiñas, J., Figueiras, A.: Effect measures in nonparametric regression with interactions between continuous exposures. Stat. Med. 25, 603–621 (2006)

    Article  MathSciNet  Google Scholar 

  3. Fan, J., Marron, J.S.: Fast implementations of nonparametric curve estimators. J. Comput. Graph. Stat. 3, 35–56 (1994)

    Article  Google Scholar 

  4. Friedman, J.H., Stuetzle, W.: Projection pursuit regression. J. Am. Stat. Assoc. 76, 817–823 (1981)

    Article  MathSciNet  Google Scholar 

  5. Härdle, W., Huet, S., Mammen, E., Sperlich, S.: Bootstrap inference in semiparametric generalized additive. Econom. Theory 20, 265–300 (2004)

    MATH  Article  Google Scholar 

  6. Hastie, T.J., Tibshirani, R.J.: Generalized Additive Models. Chapman & Hall, London (1990)

    Google Scholar 

  7. Linton, O., Sperlich, S., Van Keilegom, I.: Estimation of a semiparametric transformation model. Ann. Stat. 36, 686–718 (2008)

    MATH  Article  Google Scholar 

  8. Mammen, E., Nielsen, J.P.: Generalised structured models. Biometrika 90, 551–566 (2003)

    Article  MathSciNet  Google Scholar 

  9. Mammen, E., Linton, O., Nielsen, J.: The existence and asymptotic properties of a backfitting projection algorithm under weak conditions. Ann. Stat. 27, 1443–1490 (1999)

    MATH  MathSciNet  Google Scholar 

  10. Mammen, E., Støve, B., Tjøstheim, D.: Nonparametric additive models for panels of time series. Presentation at the International Conference on Time Series Econometrics, Finance and Risk, Perth, June 2006

  11. McCullagh, P., Nelder, J.A.: Generalized linear models. Chapman & Hall, London (1989)

    Google Scholar 

  12. Nielsen, J.P., Linton, O.B.: An optimization interpretation of integration and backfitting estimators for separable nonparametric models. J. R. Stat. Soc. B 60, 217–222 (1998)

    MATH  Article  MathSciNet  Google Scholar 

  13. Nielsen, J.P., Sperlich, S.: Smooth backfitting in practice. J. R. Stat. Soc. B 67, 43–61 (2005)

    MATH  Article  MathSciNet  Google Scholar 

  14. Opsomer, J.D.: Asymptotic properties of backfitting estimators. J. Multivar. Anal. 73, 166–179 (2000)

    MATH  Article  MathSciNet  Google Scholar 

  15. Opsomer, J.D., Ruppert, D.: Fitting a bivariate additive model by local polynomial regression. Ann. Stat. 25, 186–211 (1997)

    MATH  Article  MathSciNet  Google Scholar 

  16. Roca-Pardiñas, J., Sperlich, S.: Testing the link when the index is semiparametric—a comparison study. Comput. Stat. Data Anal. 12, 6565–6581 (2007)

    Article  Google Scholar 

  17. Roca-Pardiñas, J., Sperlich, S.: Estimating generalized structured models—a computational note. Discussion Paper, Georg August Universität Göttingen, Germany (2008)

  18. Roca-Pardiñas, J., González-Manteiga, W., Febrero-Bande, W., Prada-Sánchez, J.M., Cadarso-Suárez, C.: Predicting binary time series of SO2 using generalized additive models with unknown link function. Environmetrics 15, 1–14 (2004)

    Article  Google Scholar 

  19. Rodríguez-Poó, J.M., Sperlich, S., Vieu, P.: Semiparametric estimation of weak and strong separable models. Econom. Theory 19, 1008–1039 (2003)

    Article  Google Scholar 

  20. Rodríguez-Poó, J.M., Sperlich, S., Vieu, P.: An adaptive specification test for semiparametric models. SSRN paper, available at: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1010933 (2007)

  21. Schienle, M.: A projection approach to additive nonparametric estimation of recurrent Markov processes. Presentation at Oberwolfach Meeting, March 2007

  22. Sperlich, S., Linton, O., Härdle, W.: Integration and backfitting methods in additive models—finite sample properties and comparison. Test 8, 419–458 (1999)

    MATH  Article  Google Scholar 

  23. Sperlich, S., Tjøstheim, D., Yang, L.: Nonparametric estimation and testing of interaction in additive models. Econom. Theory 18, 197–251 (2002)

    MATH  Article  Google Scholar 

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

    MATH  Article  Google Scholar 

  25. Stone, C.J.: The dimensionality reduction principle for generalized additive models. Ann. Stat. 14, 590–606 (1986)

    MATH  Article  Google Scholar 

  26. Vilar-Fernández, J.M., Francisco-Fernández, M.: Local polynomial regression smoothers with AR-error structure. Test 11, 439–464 (2002)

    MATH  Article  MathSciNet  Google Scholar 

  27. Wahba, G., Wang, Y., Gu, C., Klein, R., Klein, B.: Smoothing spline ANOVA for exponential families, with application to the Wisconsin epidemiological study of diabetic retinopathy. Ann. Stat. 23, 1865–1895 (1995)

    MATH  Article  MathSciNet  Google Scholar 

  28. Wand, M.: Fast implementation of multivariate kernel estimators. J. Comput. Graph. Stat. 3, 433–445 (1994)

    Article  MathSciNet  Google Scholar 

  29. Wood, S.N.: Thin plate regression splines. J. R. Stat. Soc., Ser. B 65, 95–114 (2003)

    MATH  Article  Google Scholar 

  30. Yu, K., Park, B.U., Mammen, E.: Smooth backfitting in generalized additive models. Ann. Stat. 36, 228–260 (2008)

    MATH  Article  MathSciNet  Google Scholar 

  31. Xiao, Z., Linton, O., Carroll, R.J., Mammen, E.: More efficient local polynomial estimation in nonparametric regression with autocorrelated errors. J. Am. Stat. Assoc. 98, 890–992 (2003)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

  1. Departamento de Estadística e I.O., Universidad de Vigo, c/Torrecedeira 86, 36208, Vigo, Spain

    Javier Roca-Pardiñas

  2. Institut für Statistik und Ökonometrie, Georg-August Universität Göttingen, Platz der Göttinger Sieben 5, 37073, Göttingen, Germany

    Stefan Sperlich

Authors
  1. Javier Roca-Pardiñas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefan Sperlich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefan Sperlich.

Additional information

The authors thank Walter Zucchini and three anonymous referees for helpful comments and discussion. We acknowledge financial support from the Spanish Ministry of Education & Science, MTM2008-03010, and Xunta de Galicia, PGIDIT07PXIB300191PR, and the Deutsche Forschungsgemeinschaft FOR916.

Rights and permissions

Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and Permissions

About this article

Cite this article

Roca-Pardiñas, J., Sperlich, S. Feasible estimation in generalized structured models. Stat Comput 20, 367–379 (2010). https://doi.org/10.1007/s11222-009-9130-2

Download citation

Keywords

  • Generalized structured models
  • Smooth backfitting
  • Generalized varying coefficients
  • Generalized additive models
  • Computational statistics