Skip to main content
Log in

Asymptotic normality of M-estimates in the classical nonlinear regression model

  • Published:
Ukrainian Mathematical Journal Aims and scope

Sufficient conditions are obtained for the asymptotic normality of M-estimates of the unknown parameters of nonlinear regression models with discrete time and independent identically distributed errors of observations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

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. P. J. Huber, “Robust regression: asymptotics, conjectures and Monte-Carlo,” Ann. Statist., 1, No. 5, 799–821 (1973).

    Article  MATH  MathSciNet  Google Scholar 

  2. P. J. Huber, Robust Statistics, Wiley, New York (1981).

    Book  MATH  Google Scholar 

  3. F. R. Hampel, E. M. Ronchetti, P. J. Rousseeuw, and W. A. Stahel, Robust Statistics. The Approach Based on Influence Functions, Wiley, New York (1986).

    MATH  Google Scholar 

  4. A. E. Ronner, “Asymptotic normality of p-norm estimators in multiple regression,” Z. Wahrscheinlichkeitstheor. Verw. Geb., 66, 613–620 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  5. T. A. Bardadym and A. V. Ivanov, “Asymptotic normality of l α -estimates for the parameters of a nonlinear regression model,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 8, 68–70 (1988).

  6. T. O. Bardadym and O. V. Ivanov, “On the asymptotic normality of l α -estimates for the parameters of a nonlinear regression model,” Teor. Imov. Mat. Stat., Issue. 60, 1–10 (1999).

  7. X. R. Chen and Y. H. Wu, “Strong consistency of M-estimates in linear models,” J. Multivar. Anal., 27, 116–130 (1988).

    Article  MATH  Google Scholar 

  8. J. Jurečková, “Consistency of M-estimators in linear model generated by a nonmonotone and discontinuous ψ-function,” Probab. Statist., 10, 1–10 (1989).

    MATH  Google Scholar 

  9. A. V. Ivanov, “On the consistency of l α -estimates for the parameters of a regression function,” Teor. Ver. Mat. Stat., Issue 42, 42–48 (1990).

    Google Scholar 

  10. F. Liese and I. Vajda, Asymptotic Normality of M-Estimators in Nonlinear Regression, Res. Rep. UTIA, Prague, No. 1714 (1991).

    Google Scholar 

  11. F. Liese and I. Vajda, Consistency of M-Estimators in Nonlinear Regression, Res. Rept UTIA, Prague, No. 1713 (1991).

    Google Scholar 

  12. F. Liese and I. Vajda, “Consistency of M-estimates in general regression models,” J. Multivar. Anal., 50, No. 1, 93–114 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  13. F. Liese and I. Vajda, “Necessary and sufficient conditions for consistency of generalized M-estimates,” Metrika, 42, 291–324 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  14. F. Liese and I. Vajda, A General Asymptotic Theory of M-Estimators, Res. Rept. UTIA, Prague, No. 1951 (1999).

    Google Scholar 

  15. C. H. Müller, Robust Planning and Analysis of Experiments, Springer, New York (1997).

    MATH  Google Scholar 

  16. F. Liese, “Necessary and sufficient conditions for consistency of approximate M-estimators in nonlinear models,” Proc. Prague Stochast., 357–360 (1998).

  17. M. A. Arcones, “Asymptotic theory of M-estimators over a convex kernel,” Econom. Theory, 14, No. 4, 387–422 (1998).

    Article  MathSciNet  Google Scholar 

  18. Y. Wu and M. M. Zen, “A strongly consistent information criterion for linear model selection based on M-estimation,” Probab. Theory Relat. Fields, 113, 599–625 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  19. Van de S. A. Geer, Empirical Processes in M-Estimation, Cambridge University, Cambridge (2000).

    Google Scholar 

  20. I. V. Orlovs’kyi, “Consistency of the Koenker–Bassett estimates in nonlinear regression models,” Nauk. Vist. NTUU “KPI”, No. 3(35), 144–150 (2004).

  21. O. V. Ivanov and I. V. Orlovs’kyi, “Asymptotic normality of the Koenker–Bassett estimates in nonlinear regression models,” Teor. Imov. Mat. Stat., Issue 72, 30–41 (2005).

    Google Scholar 

  22. A. V. Ivanov and I. V. Orlovsky, “Parameter estimators of nonlinear quantile regression,” Theor. Stochast. Proc., 11(27), No. 3–4, 82–91 (2005).

    MathSciNet  Google Scholar 

  23. H. L. Koul, “M-estimators in linear models with long range dependent errors,” Statist. Probab. Lett., 14, 153–164 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  24. H. L. Koul, “Asymptotics of M-estimations in nonlinear regression with long-range dependence errors,” in: P. M. Robinson and M. Rosenblatt (editors), Lecture Notes in Statistics, “Proc. of the Athens Conf. on Applied Probability and Time Series Analysis,” Vol. 2, Springer, New York (1996), pp. 272–291.

    Google Scholar 

  25. H. L. Koul and K. Mukherjee, “Regression quantiles and related processes under long range dependent errors,” J. Multivar. Anal., 51, 318–337 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  26. L. Giraitis, H. L. Koul, and D. Surgailis, “Asymptotic normality of regression estimators with long memory errors,” Statist. Probab. Lett., 29, 317–335 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  27. H. L. Koul and D. Surgailis, “Asymptotic expansion of M-estimators with long memory errors,” Ann. Statist., 25, 818–850 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  28. H. L. Koul and D. Surgailis, “Second order behavior of M-estimators in linear regression with long-memory errors,” J. Statist. Planning Inference, 91, 399–412 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  29. H. L. Koul and D. Surgailis, “Robust estimators in regression models with long memory errors,” in: P. Doukhan, G. Oppenheim, and M. S. Taqqu (editors), Theory and Application of Long-Range Dependence, Birkhäuser, Boston (2003), pp. 339–353.

    Google Scholar 

  30. L. Giraitis and H. L. Koul, “Estimation of the dependence parameter in linear regression with long-range dependent errors,” Statist. Probab. Lett., 29, 317–335 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  31. H. L. Koul, R. T. Baillie, and D. Surgailis, “Regression model fitting with a long memory covariance process,” Economic Theory, 20, 485–512 (2004).

    MATH  MathSciNet  Google Scholar 

  32. A. V. Ivanov and N. N. Leonenko, “Asymptotic behavior of M-estimators in continuous-time nonlinear regression with long-range dependent errors,” Random Oper. Stochast. Equat., 10, No. 3, 201–222 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  33. A. V. Ivanov and I. V. Orlovsky, “L p -estimates in nonlinear regression with long-range dependence,” Theor. Stochast. Proc., 7, No. 3–4, 38–49 (2002).

    Google Scholar 

  34. O. V. Ivanov and I. V. Orlovs’kyi, “Consistency of M-estimates in nonlinear regression models with continuum time,” Nauk. Vist. NTUU “KPI”, No. 4, 140–147 (2005).

  35. I. V. Orlovsky, “M-estimates in nonlinear regression with weak dependence,” Theor. Stochast. Proc., 9, No. 1–2, 108–122 (2003).

    MathSciNet  Google Scholar 

  36. I. V. Orlovs’kyi, Asymptotic Properties ofM-Estimates for the Parameters of Nonlinear Regression Models [in Ukrainian], Candidate-Degree Thesis (Physics and Mathematics), Kyiv (2007).

  37. R. I. Jennrich, “Asymptotic properties of nonlinear least squares estimators,” Ann. Math. Statist., 40, 633–643 (1969).

    Article  MATH  MathSciNet  Google Scholar 

  38. J. Pfanzagl, “On the measurability and consistency of minimum contrast estimates,” Metrika, 14, 249–272 (1969).

    Article  MATH  Google Scholar 

  39. L. Schmetterer, Einführung in die Mathematische Statistik, Springer, New York (1966).

    MATH  Google Scholar 

  40. A. V. Ivanov, Asymptotic Theory of Nonlinear Regression, Kluwer, Dordrecht (1997).

    MATH  Google Scholar 

  41. R. N. Bhattacharya and R. R. Rao, Normal Approximation and Asymptotic Expansions, Wiley, New York (1976).

    MATH  Google Scholar 

  42. J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford (1965).

    MATH  Google Scholar 

  43. Yu. V. Goncharenko and S. I. Lyashko, Brouwer Theorem [in Russian], KII, Kiev (2000).

    Google Scholar 

  44. I. I. Gikhman, A. V. Skorokhod, and M. I. Yadrenko, Probability Theory and Mathematical Statistics [in Russian], Vyshcha Shkola, Kiev (1988).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 11, pp. 1470–1488, November, 2008.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ivanov, O.V., Orlovs’kyi, I.V. Asymptotic normality of M-estimates in the classical nonlinear regression model. Ukr Math J 60, 1716–1739 (2008). https://doi.org/10.1007/s11253-009-0165-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-009-0165-5

Keywords

Navigation