Theory of Probability and Mathematical Statistics
Asymptotic properties of Ibragimov's estimator for a parameter of the spectral density of the random noise in a nonlinear regression model
A. V. Ivanov, V. V. Prikhod’ko
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Abstract: A nonlinear regression model with continuous time is considered. The consistency and asymptotic normality of the Ibragimov estimator for a parameter of the spectral density of the Gaussian stationary noise are obtained.
Keywords: Nonlinear regression model with continuous time, Gaussian stationary noise, residual periodogram, Ibragimov's estimator of a spectral density, consistency, asymptotic normality
Bibliography: 1. A. V. Ivanov, Asymptotic Theory of Nonlinear Regression, Kluwer Academic Publishers Group, Dordrecht-Boston-London, 1997.
2. O. V. Ivanov and K. K. Moskvichova, Stochastic asymptotic expansion of correlogram estimator of the correlation function of random noise in nonlinear regression model, Teor. Ĭmovir. Mat. Stat. 90 (2014), 77-90; English transl. in Theory Probab. Math. Statist. 90 (2015), 87-101.
3. O. V. Ivanov and K. K. Moskvichova, Asymptotic expansion of the moments of correlogram estimator for the random-noise covariance function in the nonlinear regression model, Ukrain. Mat. Zh. 66 (2014), no. 6, 787-805; English transl. in Ukrainian Math. J. 66 (2014), no. 6, 884-904.
4. H. L. Koul and D. Surgailis, Asymptotic normality of the Wittle estimator in linear regression models with long memory errors, Stat. Inference Stoch. Process. (2000), no. 3, 129-147.
5. O. V. Ivanov and V. V. Prykhod'ko, On the Whittle estimator of the parameter of spectral density of random noise in the nonlinear regression model, Ukrain. Mat. Zh. 67 (2015), no. 8, 1050-1067; English transl. in Ukrainian Math. J. 67 (2016), no. 8, 1183-1203.
6. A. V. Ivanov and N. N. Leonenko, Semiparametric analysis of long-range dependence in nonlinear regression, J. Statist. Plann. Inference (2008), no. 138, 1733-1753.
7. 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. Plann. Inference (2004), no. 123, 161-185.
8. N. N. Leonenko and L. M. Sakhno, On the Whittle estimators for some classes of continuous-parameter random processes and fields, Statistic Probab. Lett. (2006), no. 76, 781-795.
9. I. A. Ibragimov, On maximum likelihood estimation of parameters of the spectral density of stationary time series, Teor. Veroyatnost. Primenen. 12 (1967), no. 1, 128-134; English transl. in Theory Probab. Appl. 12 (1967), no. 1, 115-119.
10. N. N. Leonenko and E. M. Moldavs'ka, Minimum contrast estimators of a parameter of the spectral density of continuous time random fields, Teor. Ĭmovir. Mat. Stat. (1998), no. 58, 92-103; English transl. in Theor. Probab. Math. Statist. (1999), no. 58, 101-112.
11. C. R. Rao, Linear Statistical Inference and its Applications, John Wiley & Sons, New York-London-Sydney, 1965.
12. N. N. Leonenko and A. V. Ivanov, Statistical Analysis of Random Fields, Vyshcha Shkola'', Kiev, 1986; English transl., Kluwer Academic Publishers Group, Dordrecht, 1989.
13. A. V. Ivanov, N. N. Leonenko, M. D. Ruiz-Medina, and I. N. Savich, Limit theorems for weighted nonlinear transformation of Gaussian stationary processes with singular spectra, Ann. Probab. 41 (2013), no. 2, 1088-1114.
14. A. V. Ivanov, N. N. Leonenko, M. D. Ruiz-Medina and B. M. Zhurakovsky, Estimation of harmonic component in regression with cyclically dependent errors, Statistics 41 (2015), no. 1, 156-186.
15. N. I. Achiezer, Theory of Approximation, ''Nauka'', Moscow, 1965; English transl., Dover Publications, Inc., New York, 1992.
16. I. A. Ibragimov, On estimation of the spectral function of a stationary Gaussian process, Teor. Veroyatnost. Primenen. 8 (1963), no. 4, 391-430; English transl. in Theory Probab. Appl. 8 (1963), no. 4, 366-401.
17. R. Bentkus, The error in estimating the spectral function of a stationary process, Litovsk. Mat. Sb. 1 (1972), no. XII, 55-71. (Russian)
18. R. M. Espejo, N. N.Leonenko, A. Olenko, and M. D. Ruiz-Medina, On a class of minimum contrast estimators for Gegenbauer random fields, Test 24 (2015), no. 4, 657-680.
19. V. V. Anh, J. M. Angulo, and M. D. Ruiz-Medina, Possible long-range dependence in fractional random field, J. Statist. Plan. Inference 80 (1999), no. 1/2, 95-110.
20. V. V. Anh, N. N. Leonenko, and R. McVinish, Models for fractional Riesz-Bessel motion and related processes, Fractals 3 (2001), no. 9, 329-346.
21. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 4th edition, ''Nauka'', Moscow, 1963; English transl., Academic Press, New York-London, 1965.
