Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Asymptotic Properties of the M-Estimates of Parameters in a Nonlinear Regression Model with Discrete Time and Singular Spectrum

  • 20 Accesses

We study nonlinear regression models with discrete time and errors of observations whose spectrum is singular. Sufficient conditions are established for the consistency, asymptotic uniqueness, and asymptotic normality of the M-estimates of unknown parameters.

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

References

  1. 1.

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

  2. 2.

    H. L. Koul, “Asymptotics of M-estimations in non-linear regression with long-range dependence errors,” in: Proc. of the Athens Conference on Applied Probability and Time Series Analysis, Vol. II, Springer, New York (1996), pp. 272–291.

  3. 3.

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

  4. 4.

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

  5. 5.

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

  6. 6.

    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).

  7. 7.

    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 Applications of Long-Range Dependence, Birkhäuser, Boston (2003), pp. 339–353.

  8. 8.

    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).

  9. 9.

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

  10. 10.

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

  11. 11.

    A. V. Ivanov and N. N. Leonenko, “Robust estimators in nonlinear regression models with long-range dependence,” in: L. Pronzato and A. Zhigljavsky (editors), Optimal Design and Related Areas in Optimization and Statistics, Springer, Berlin (2009), pp. 193–221.

  12. 12.

    A.V. Ivanov, “Asymptotic properties of L p -estimators,” Theory Stochast. Proc., 14(30), No. 1, 60–68 (2008).

  13. 13.

    A.V. Ivanov and I. V. Orlovsky, “L p -estimates in nonlinear regression with long-range dependence,” Theory Stochast. Proc., 7(23), No. 3-4, 38–49 (2002).

  14. 14.

    O. V. Ivanov and I. V. Orlovs’kyi, “Consistency of M-estimates in nonlinear regression models with continuous time,” Nauk. Visti Nats. Tekh. Univ. Ukr. “Kyiv. Politekh. Inst.”, No. 4(42), 140–147 (2005).

  15. 15.

    O. V. Ivanov and I. V. Orlovs’kyi, “On the uniqueness of M-estimates for parameters of nonlinear regression models,” Nauk. Visti Nats. Tekh. Univ. Ukr. “Kyiv. Politekh. Inst.”, No. 4(46), 135–141 (2009).

  16. 16.

    O. V. Ivanov and I. V. Orlovs’kyi, “Asymptotic properties of M-estimates for the parameters of nonlinear regression models with random noise whose spectrum is singular,” Teor. Imovirn. Mat. Statist., Issue 93, 34–49 (2015).

  17. 17.

    I. M. Savych, “Consistency of quantile estimates in regression models with strongly dependent noise,” Teor. Imovirn. Mat. Statist., Issue 82, 128–136 (2010).

  18. 18.

    I. V. Orlovsky, “M-estimates in nonlinear regression with weak dependence,” Theory Stochast. Proc., 9(25), No. 1-2, 108–122 (2003).

  19. 19.

    A.V. Ivanov and I. V. Orlovsky, “Consistency ofM-estimates in general nonlinear model,” Theory Stochast. Proc., 13(29), No. 1-2, 86–97 (2007).

  20. 20.

    A.V. Ivanov, N. N. Leonenko, M. D. Ruiz-Medina, and I. N. Savych, “Limit theorems for weighted nonlinear transformations of Gaussian processes with singular spectra,” Ann. Probab., 41, No. 2, 1088–1114 (2013).

  21. 21.

    Yu. V. Goncharenko and S. I. Lyashko, Brauer Theorem [in Russian], Kii, Kiev (2000).

  22. 22.

    V. V. Anh, V. P. Knopova, and N. N. Leonenko, “Continuous-time stochastic processes with cyclical long-range dependence,” Aust. NZ J. Statist., 46, 275–296 (2004).

  23. 23.

    A.V. Ivanov, N. N. Leonenko, M. D. Ruiz-Medina, and B. M. Zhurakovsky, “Estimation of harmonic component in regression with cyclically dependent errors,” Statist.: J. Theor. Appl. Statist., 49, No. 1, 156–186 (2015).

  24. 24.

    E. J. Hannan, Multiple Time Series, Wiley, New York (1970).

  25. 25.

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

  26. 26.

    L. Schmetterer, Einführung in die Mathematische Statistik [Russian translation], Nauka, Moscow (1976).

  27. 27.

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

  28. 28.

    U. Grenander, “On the estimation of regression coefficients in the case of an autocorrelated disturbance,” Ann. Statist., 25, No. 2, 252–272 (1954).

  29. 29.

    I. A. Ibragimov and Yu. A. Rozanov, Gaussian Random Processes [in Russian], Nauka, Moscow (1970).

  30. 30.

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

  31. 31.

    O. V. Ivanov and I. M. Savych, “μ-Admissibility of the spectral density of a strongly dependent random noise in nonlinear regression models,” Nauk. Visti Nats. Tekh. Univ. Ukr. “Kyiv. Politekh. Inst.”, No. 1, 143–148 (2009).

  32. 32.

    N. N. Leonenko and A.V. Ivanov, Statistical Analysis of Random Fields [in Russian], Vyshcha Shkola, Kiev (1986).

  33. 33.

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

  34. 34.

    J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford (1982).

Download references

Author information

Correspondence to O. V. Ivanov.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 1, pp. 28–51, January, 2017.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ivanov, O.V., Orlovs’kyi, I.V. Asymptotic Properties of the M-Estimates of Parameters in a Nonlinear Regression Model with Discrete Time and Singular Spectrum. Ukr Math J 69, 32–61 (2017). https://doi.org/10.1007/s11253-017-1346-2

Download citation