Skip to main content
Log in

On the Whittle Estimator of the Parameter of Spectral Density of Random Noise in the Nonlinear Regression Model

  • Published:
Ukrainian Mathematical Journal Aims and scope

We consider a nonlinear regression model with continuous time and establish the consistency and asymptotic normality of the Whittle minimum contrast estimator for the parameter of spectral density of stationary Gaussian noise.

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. A.V. Ivanov and N. N. Leonenko, Statistical Analysis of Random Fields, Kluwer AP, Dordrecht (1989).

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  3. P. Whittle, Hypothesis Testing in Time Series, Hafner, New York (1951).

    MATH  Google Scholar 

  4. P. Whittle, “Estimation and information in stationary time series,” Ark. Mat., No. 2, 423–434 (1953).

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

    Book  MATH  Google Scholar 

  6. E. J. Hannan, “The asymptotic theory of linear time series models,” J. Appl. Probab., No. 10, 130–145 (1973).

  7. W. Dunsmuir and E. J. Hannan, “Vector linear time series models,” Adv. Appl. Probab., No. 8, 339–360 (1976).

  8. X. Guyon, “Parameter estimation for a stationary process on a d-dimensional lattice,” Biometrica, No. 69, 95–102 (1982).

  9. M. R. Rosenblatt, Stationary Sequences and Random Fields, Birkhäuser, Boston (1985).

    Book  MATH  Google Scholar 

  10. R. Fox and M. S. Taqqu, “Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series,” Ann. Statist., 2, No. 14, 517–532 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  11. R. Yu. Bentkus and R. R. Malinkevicius, “Statistical estimation of a multidimensional parameter of a spectral density. I,” Lith. Math. J., 2, No. 28, 115–126 (1988).

    MathSciNet  Google Scholar 

  12. R. Yu. Bentkus and R. R. Malinkevicius, “Statistical estimation of a multidimensional parameter of a spectral density. II,” Lith. Math. J., 3, No. 28, 209–221 (1988).

    MathSciNet  Google Scholar 

  13. R. Dahlhaus, “Efficient parameter estimation for self-similar processes,” Ann. Statist., 17, 1749–1766 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  14. C. Heyde and R. Gay, “On asymptotic quasi-likelihood stochastic process,” Stochast. Process. Appl., No. 31, 223–236 (1989).

  15. L. Giraitis and D. Surgailis, “A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotic normality of Whittle estimate,” Probab. Theory Relat. Fields, No. 86, 87–104 (1990).

  16. C. Heyde and R. Gay, “Smoothed periodogram asymptotic and estimation for processes and fields with possible long-range dependence,” Stochast. Process. Appl., No. 45, 169–182 (1993).

  17. L. Giraitis and M. S. Taqqu, “Whittle estimator for finite-variance non-Gaussian time series with long memory,” Ann. Statist., 1, No. 27, 178–203 (1999).

    MathSciNet  MATH  Google Scholar 

  18. J. Gao, N. N. Anh, C. C. Heyde, and Q. Tieng, “Parameter estimation of stochastic processes with long-range dependence and intermittency,” J. Time Ser. Anal., No. 22, 517–535 (2001).

  19. J. Gao, “Modelling long-range-dependent Gaussian processes with application in continuous-time financial models,” J. Appl. Probab., No. 41, 467–485 (2004).

  20. N. N. Leonenko and L. M. Sakhno, “On the Whittle estimators for some classes of continuous-parameter random processes and fields,” Statist. Probab. Lett., No. 76, 781–795 (2006).

  21. L. M. Sakhno, Asymptotic and Spectral Methods for the Statistical Estimation of Random Processes and Fields [in Ukrainian], Doctoral-Degree Thesis (Physics and Mathematics), Kyiv (2012).

  22. V. V. Anh, N. N. Leonenko, and L. M. Sakhno, “On a class of minimum contract estimators for fractional stochastic processes and fields,” J. Statist. Planning Inference, No. 123, 161–185 (2004).

  23. H. L. Koul and D. Surgailis, “Asymptotic normality of the Whittle estimator in linear regression models with long memory errors,” Statist. Inference Stochast. Processes, No. 3, 129–147 (2000).

  24. N. I. Akhiezer, Lectures on Approximation Theory [in Russian], Nauka, Moscow (1965).

    Google Scholar 

  25. I. A. Ibragimov, “On the estimation of the spectral function of a stationary Gaussian process,” Teor. Ver. Primen., 4, No. 8, 391–430 (1963).

    Google Scholar 

  26. R. Bentkus, “On the error of the estimator of spectral function for a stationary process,” Lit. Mat. Sb., 1, No. 12, 55–71 (1972).

    MathSciNet  Google Scholar 

  27. Yu. A. Rozanov, Stationary Random Processes [in Russian], Fizmatgiz, Moscow (1963).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 8, pp. 1050–1067, August, 2015.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ivanov, O.V., Prykhod’ko, V.V. On the Whittle Estimator of the Parameter of Spectral Density of Random Noise in the Nonlinear Regression Model. Ukr Math J 67, 1183–1203 (2016). https://doi.org/10.1007/s11253-016-1145-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-016-1145-1

Keywords

Navigation