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Estimating the innovation distribution in nonparametric autoregression
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  • Published: 09 February 2008

Estimating the innovation distribution in nonparametric autoregression

  • Ursula U. Müller1,
  • Anton Schick2 &
  • Wolfgang Wefelmeyer3 

Probability Theory and Related Fields volume 144, pages 53–77 (2009)Cite this article

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Abstract

We prove a Bahadur representation for a residual-based estimator of the innovation distribution function in a nonparametric autoregressive model. The residuals are based on a local linear smoother for the autoregression function. Our result implies a functional central limit theorem for the residual-based estimator.

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References

  1. An H.Z. and Huang F.C. (1996). The geometrical ergodicity of nonlinear autoregressive models. Statist. Sinica 6: 943–956

    MATH  MathSciNet  Google Scholar 

  2. Akritas M.G. and Van Keilegom I. (2001). Non-parametric estimation of the residual distribution. Scand. J. Statist. 28: 549–567

    Article  MATH  MathSciNet  Google Scholar 

  3. Berkes, I., Horváth, L.: Empirical processes of residuals. In: Dehling, H., Mikosch, T., Sørensen, M. (eds.) Empirical Process Techniques for Dependent Data, pp. 195–209, Birkhäuser, Boston (2002)

  4. Berkes I. and Horváth L. (2003). Limit results for the empirical process of squared residuals in GARCH models. Stoch. Process. Appl. 105: 271–298

    Article  MATH  Google Scholar 

  5. Bhattacharya R.N. and Lee C. (1995). Ergodicity of nonlinear first order autoregressive models. J. Theoret. Probab. 8: 207–219

    Article  MATH  MathSciNet  Google Scholar 

  6. Bhattacharya, R., Lee, C.: On geometric ergodicity of nonlinear autoregressive models. Statist. Probab. Lett. 22, 311–315 (1995b). Erratum: 41 439–440 (1999)

    Google Scholar 

  7. Billingsley P. (1968). Convergence of Probability Measures. Wiley, New York

    MATH  Google Scholar 

  8. Boldin M.V. (1982). Estimation of the distribution of noise in an autoregression scheme. Theory Probab. Appl. 27: 866–871

    Article  MATH  MathSciNet  Google Scholar 

  9. Boldin M.V. (1998). On residual empirical distribution functions in ARCH models with applications to testing and estimation. Mitt. Math. Sem. Giessen 235: 49–66

    MATH  MathSciNet  Google Scholar 

  10. Cheng F. (2005). Asymptotic distributions of error density and distribution function estimators in nonparametric regression. J. Statist. Plann. Inference 128: 327–349

    Article  MATH  MathSciNet  Google Scholar 

  11. Dette H., Neumeyer N. and Van Keilegom I. (2007). A new test for the parametric form of the variance function in nonparametric regression. J. Roy. Statist. Soc. Ser. B 69: 903–917

    Article  MathSciNet  Google Scholar 

  12. Durbin J. (1973). Weak convergence of the sample distribution function when parameters are estimated. Ann. Statist. 1: 279–290

    Article  MATH  MathSciNet  Google Scholar 

  13. Einmahl J. and Van Keilegom I. (2008). Specification tests in nonparametric regression. J. Econom. 143: 88–102

    Article  MathSciNet  Google Scholar 

  14. Freedman D.A. (1975). On tail probabilities for martingales. Ann. Probab. 3: 100–118

    Article  MATH  Google Scholar 

  15. Gill R.D. (1989). Non- and semi-parametric maximum likelihood estimators and the von Mises method. I. With a discussion by J. A. Wellner and J. Præstgaard and a reply by the author. Scand. J. Statist. 16: 97–128

    MATH  MathSciNet  Google Scholar 

  16. Grama I.G. and Neumann M.H. (2006). Asymptotic equivalence of nonparametric autoregression and nonparametric regression. Ann. Statist. 34: 1701–1732

    Article  MathSciNet  Google Scholar 

  17. Hansen, B.E.: Uniform convergence rates for kernel estimation with dependent data. To appear in: Econom. Theory 24, (2008)

  18. Horváth L., Kokoszka P. and Teyssière G. (2001). Empirical process of the squared residuals of an ARCH sequence. Ann. Statist. 29: 445–469

    Article  MATH  MathSciNet  Google Scholar 

  19. Kawczak J., Kulperger R. and Yu H. (2005). The empirical distribution function and partial sum process of residuals from a stationary ARCH with drift process. Ann. Inst. Statist. Math. 57: 747–765

    Article  MATH  MathSciNet  Google Scholar 

  20. Kiwitt, S., Nagel, E.-R., Neumeyer, N.: Empirical likelihood estimators for the error distribution in nonparametric regression models. Technical Report, Faculty of Mathematics, University of Bochum (2005)

  21. Koul H.L. (1969). Asymptotic behavior of Wilcoxon type confidence regions in multiple linear regression. Ann. Math. Statist. 40: 1950–1979

    Article  MATH  MathSciNet  Google Scholar 

  22. Koul H.L. (1970). Some convergence theorems for ranks and weighted empirical cumulatives. Ann. Math. Statist. 41: 1768–1773

    Article  MATH  MathSciNet  Google Scholar 

  23. Koul H.L. (1991). A weak convergence result useful in robust autoregression. J. Statist. Plann. Inference 29: 291–308

    Article  MATH  MathSciNet  Google Scholar 

  24. Koul H.L. (2002). Weighted Empirical Processes in Dynamic Nonlinear Models. Lecture Notes in Statistics 166. Springer, New York

    Google Scholar 

  25. Koul H.L. and Ling S. (2006). Fitting an error distribution in some heteroscedastic time series models. Ann. Statist. 34: 994–1012

    Article  MATH  MathSciNet  Google Scholar 

  26. Koul H.L. and Ossiander M. (1994). Weak convergence of randomly weighted dependent residual empiricals with applications to autoregression. Ann. Statist. 22: 540–562

    Article  MATH  MathSciNet  Google Scholar 

  27. Kulperger R. and Yu H. (2005). High moment partial sum processes of residuals in GARCH models and their applications. Ann. Statist. 33: 2395–2422

    Article  MATH  MathSciNet  Google Scholar 

  28. Lee S. and Taniguchi M. (2005). Asymptotic theory for ARCH-SM models: LAN and residual empirical processes. Statist. Sinica 15: 215–234

    MATH  MathSciNet  Google Scholar 

  29. Loynes R.M. (1980). The empirical distribution function of residuals from generalised regression. Ann. Statist. 8: 285–299

    Article  MATH  MathSciNet  Google Scholar 

  30. Mammen E. (1996). Empirical process of residuals for high-dimensional linear models. Ann. Statist. 24: 307–335

    Article  MATH  MathSciNet  Google Scholar 

  31. Müller U.U., Schick A. and Wefelmeyer W. (2004). Estimating linear functionals of the error distribution in nonparametric regression. J. Statist. Plann. Inference 119: 75–93

    Article  MATH  MathSciNet  Google Scholar 

  32. Müller U.U., Schick A. and Wefelmeyer W. (2007). Estimating the error distribution function in semiparametric regression. Statist. Decisions 25: 1–18

    Article  MATH  MathSciNet  Google Scholar 

  33. Neumeyer N. and Dette H. (2005). A note on one-sided nonparametric analysis of covariance by ranking residuals. Math. Methods Statist. 14: 80–104

    MathSciNet  Google Scholar 

  34. Pardo-Fernández J.C., Van Keilegom I. and González-Manteiga W. (2007). Testing for the equality of k regression curves. Statist. Sinica 17: 1115–1137

    MATH  MathSciNet  Google Scholar 

  35. Portnoy S. (1986). Asymptotic behavior of the empiric distribution of M-estimated residuals from a regression model with many parameters. Ann. Statist. 14: 1152–1170

    Article  MATH  MathSciNet  Google Scholar 

  36. Shorack G.R. (1984). Empirical and rank processes of observations and residuals. Canad. J. Statist. 12: 319–332

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  38. Van Keilegom, I., González Manteiga, W., Sánchez Sellero, C.: Goodness-of-fit tests in parametric regression based on the estimation of the error distribution. To appear in: Test (2007)

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Author information

Authors and Affiliations

  1. Department of Statistics, Texas A&M University, College Station, TX, 77843-3143, USA

    Ursula U. Müller

  2. Department of Mathematical Sciences, Binghamton University, Binghamton, NY, 13902-6000, USA

    Anton Schick

  3. Mathematical Institute, University of Cologne, Weyertal 86-90, 50931, Cologne, Germany

    Wolfgang Wefelmeyer

Authors
  1. Ursula U. Müller
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  2. Anton Schick
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  3. Wolfgang Wefelmeyer
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Corresponding author

Correspondence to Anton Schick.

Additional information

The research of A. Schick was supported in part by NSF Grant DMS0405791.

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Cite this article

Müller, U.U., Schick, A. & Wefelmeyer, W. Estimating the innovation distribution in nonparametric autoregression. Probab. Theory Relat. Fields 144, 53–77 (2009). https://doi.org/10.1007/s00440-008-0141-2

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  • Received: 04 July 2007

  • Revised: 16 January 2008

  • Published: 09 February 2008

  • Issue Date: May 2009

  • DOI: https://doi.org/10.1007/s00440-008-0141-2

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Keywords

  • Residual-based empirical distribution function
  • Local linear smoother
  • Bahadur representation

Mathematics Subject Classification (2000)

  • 62M05
  • 62M10
  • 62G30
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