Skip to main content
SpringerLink
Log in

Empirical likelihood estimators for the error distribution in nonparametric regression models

  • Published:
Mathematical Methods of Statistics Aims and scope Submit manuscript

Abstract

The aim of this paper is to show that existing estimators for the error distribution in non-parametric regression models can be improved when additional information about the distribution is included by the empirical likelihood method. The weak convergence of the resulting new estimator to a Gaussian process is shown and the performance is investigated by comparison of asymptotic mean squared errors and by means of 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

Similar content being viewed by others

References

  1. M. Akritas and I. Van Keilegom, “Nonparametric Estimation of the Residual Distribution”, Scand. J. Statist. 28, 549–567 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  2. H. Bonnal and E. Renault, On the Efficient Use of the Informational Content of Estimating Equations: Implied Probabilities and Euclidean Empirical Likelihood. Technical report (2004). http://ideas.repec.org/p/cir/cirwor/2004s-18.html

  3. F. Cheng, “Weak and Strong Uniform Consistency of a Kernel Error Density Estimator in Nonparametric Regression”, J. Statist. Plann. Inference 119, 95–107 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  4. F. Cheng, “Consistency of Error Density and Distribution Function Estimators in Nonparametric Regression”, Statist. Probab. Lett. 59, 257–270 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  5. H. Dette, N. Neumeyer, and I. Van Keilegom, “A New Test for the Parametric Form of the Variance Function in Nonparametric Regression”, J. Roy. Statist. Soc. Ser. B, 69, 903–917 (2007).

    Article  Google Scholar 

  6. M. D. Donsker, “Justification and Extension of Doob’s Heuristic Approach to the Kolmogorov-Smirnov Theorems”, Ann. Math. Statist. 23, 277–281 (1952).

    Article  MATH  MathSciNet  Google Scholar 

  7. T. DiCiccio and J. P. Romano, “On Adjustments Based on the Signed Root of the Empirical Likelihood Ratio Statistic”, Biometrika 76, 447–456 (1989).

    MATH  MathSciNet  Google Scholar 

  8. T. DiCiccio, P. Hall, and J. P. Romano, “Comparison of Parametric and Empirical Likelihood Functions”, Biometrika 76, 465–476 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  9. T. DiCiccio, P. Hall, and J. P. Romano, “Empirical Likelihood is Bartlett-Correctable”, Ann. Statist. 19, 1053–1061 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  10. J. H. J. Einmahl and I. W. McKeague, “Empirical Likelihood Based Hypothesis Testing”, Bernoulli 9, 267–290 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  11. M. Genz, Anwendungen des Empirischen Likelihood-Schätzers der Fehlerverteilung in AR(1)-Prozessen, Dissertation, Justus-Liebig-Universität Giessen (2004) [in German], http://geb.uni-giessen.de/geb/volltexte/2005/2069/pdf/GenzMichael-2004-12-03.pdf

  12. M. Genz and E. Haeusler, “Empirical Processes with Estimated Parameters under Auxiliary Information”, J. Comput. Appl. Math. 186, 191–216 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  13. P. Hall and B. LaScala, “Methodology and Algorithms of Empirical Likelihood”, Internat. Statist. Rev. 58, 109–127 (1990).

    Article  MATH  Google Scholar 

  14. P. Hall, “Pseudo-Likelihood Theory for Empirical Likelihood”, Ann. Statist. 18, 121–140 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  15. Y. Kitamura, “Empirical Likelihood Methods with Weakly Dependent Processes”, Ann. Statist. 25, 2084–2102 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  16. Y. Kitamura, “Asymptotic Optimality of Empirical Likelihood for Testing Moment Restrictions”, Econometrica 69, 1661–1672 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  17. Y. Kitamura, G. Tripathi, H. Ahn, “Empirical Likelihood-Based Inference in Conditional Moment Restriction Models”, Econometrica 72, 1667–1714 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  18. S. Kiwitt, E.-R. Nagel, and N. Neumeyer, Empirical Likelihood Estimators for the Error Distribution in Nonparametric Regression Models, Technical report, available from the authors on request (2007).

  19. H. L. Koul, Weighted Empirical Processes in Dynamic Nonlinear Models, 2nd ed. (Springer, New York, 2002).

    MATH  Google Scholar 

  20. U. Mueller, A. Schick, and W. Wefelmeyer, “Estimating Linear Functionals of the Error Distribution in Nonparametric Regression”, J. Statist. Planning Inf. 119, 75–93 (2004).

    Article  MATH  Google Scholar 

  21. U. Mueller, A. Schick, and W. Wefelmeyer, “Estimating Functionals of the Error Distribution in Parametric and Nonparametric Regression”, J. Nonparam. Statist. 16, 525–548 (2004).

    Article  MATH  Google Scholar 

  22. U. Mueller, A. Schick, and W. Wefelmeyer, “Estimating the Error Distribution Function in Semiparametric Regression”, Statist. Decisions 25, 1–18 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  23. E. A. Nadaraya, “On Nonparametric Estimates of Density Functions and Regression Curves”, Theory Probab. Appl. 10, 186–190 (1964).

    Article  Google Scholar 

  24. E.-R. Nagel, Der Empirical-Likelihood-Schätzung für die Fehlerverteilung im linearen Regressions-modell, E. Häusler, supervisor (Justus Libig Universität, Giessen, 2002) [in German].

    Google Scholar 

  25. E.-R. Nagel, Empirical-Likelihood-Schätzung der Fehlerverteilung im nichtparametrischen Regressionsmodell, PhD. Dissertation, Ruhr-Universität Bochum (2006) [in German], http://deposit.ddb.de/cgi-bin/dokserv?idn=983354553.

  26. N. Neumeyer, “Smooth Residual Bootstrap for Empirical Processes of Nonparametric Regression Residuals”, Scand. J. Statist. (2008) (in press).

  27. N. Neumeyer and H. Dette, “A Note on One-Sided Nonparametric Analysis of Covariance by Ranking Residuals”, Math. Methods Statist. 14, 80–104 (2005).

    MathSciNet  Google Scholar 

  28. N. Neumeyer, H. Dette, and E.-R. Nagel, “Bootstrap Tests for the Error Distribution in Linear and Nonparametric Regression Models”, Austral. New Zealand J. Statist. 48, 129–156 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  29. A. B. Owen, “Empirical Likelihood Ratio Confidence Intervals for a Single Functional”, Biometrika 75(2), 237–249 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  30. A. B. Owen, Empirical Likelihood (Chapman and Hall/CRC, 2001).

  31. J. C. Pardo-Fernández, I. Van Keilegom, and W. González-Manteiga, “Goodness-of-Fit Tests for Parametric Models in Censored Regression”, Canad. J. Statist. 35, 249–264 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  32. W. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++, The Art of Scientific Computing, 2nd ed. (Cambridge Univ. Press, Cambridge, 2002).

    Google Scholar 

  33. J. Qin and J. Lawless, “Empirical Likelihood and General Estimating Equations”, Ann. Statist. 22, 300–325 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  34. G. Qin, “Consistent Residuals Density Estimation in Nonparametric Regression Under m(n)-Dependent Sample”, J. Math. Research and Exposition 16, 5005–5516 (1996).

    Google Scholar 

  35. G. Qin, S. Shi, and G. Chai, “Asymptotics of the Residual Density Estimation in Nonparametric Regression Under m(n)-Dependence”, Appl. Math., A Journal of Chinese Universities Ser. B 11, 59–76 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  36. A. W. van der Vaart, Asymptotic Statistics (Cambridge Univ. Press, Cambridge, 1998).

    MATH  Google Scholar 

  37. A. W. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes (Springer, New York, 1996).

    MATH  Google Scholar 

  38. I. Van Keilegom, W. González-Manteiga, and C. Sánchez Sellero, “Goodness-of-Fit Tests in Parametric Regression Based on Estimating the Error Distribution”, Test 17, 401–415 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  39. G. S. Watson, “Smooth Regression Analysis”, Sankhya A 26, 359–372 (1964).

    MATH  Google Scholar 

  40. B. Zhang, “Estimating a Distribution Function in the Presence of Auxiliary Information”, Metrika 46, 221–244 (1997).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universität Hamburg, Hamburg, Germany

    S. Kiwitt & N. Neumeyer

  2. Ruhr-Universität, Bochum, Germany

    E. -R. Nagel

Authors
  1. S. Kiwitt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. -R. Nagel
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. N. Neumeyer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. Neumeyer.

About this article

Cite this article

Kiwitt, S., Nagel, E.R. & Neumeyer, N. Empirical likelihood estimators for the error distribution in nonparametric regression models. Math. Meth. Stat. 17, 241–260 (2008). https://doi.org/10.3103/S1066530708030058

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066530708030058

Key words

2000 Mathematics Subject Classification

Search

Navigation