Consistency of the least squares estimator of the amplitudes and angular frequencies of a sum of harmonic oscillations in models with long-range dependence
Author:
A. V. Ivanov
Translated by:
Oleg Klesov
Journal:
Theor. Probability and Math. Statist. 80 (2010), 61-69
MSC (2000):
Primary 62J02; Secondary 62J99
DOI:
https://doi.org/10.1090/S0094-9000-2010-00794-0
Published electronically:
August 18, 2010
MathSciNet review:
2541952
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The strong consistency of least squares estimators of unknown amplitudes and angular frequencies of the sum of harmonic oscillations observed in a strongly dependent Gaussian stationary noise is proved in the paper.
References
- M. G. Serebrennikov and A. A. Pervozvanskiĭ, The Detection of Latent Periodicities, Nauka, Moscow, 1965. (Russian)
- P. Whittle, The simultaneous estimation of a time series harmonic components and covariance structure, Trabajos Estadíst. 3 (1952), 43–57 (English, with Spanish summary). MR 51487
- A. M. Walker, On the estimation of a harmonic component in a time series with stationary dependent residuals, Advances in Appl. Probability 5 (1973), 217–241. MR 336943, DOI https://doi.org/10.2307/1426034
- E. J. Hannan, The estimation of frequency, J. Appl. Probability 10 (1973), 510–519. MR 370977, DOI https://doi.org/10.2307/3212772
- A. V. Ivanov, A solution of the problem of detecting hidden periodicities, Teor. Veroyatnost. i Mat. Statist. 20 (1979), 44–59, 156–157 (Russian, with English summary). MR 529259
- Alexander V. Ivanov, Asymptotic theory of nonlinear regression, Mathematics and its Applications, vol. 389, Kluwer Academic Publishers Group, Dordrecht, 1997. MR 1472234
- P. S. Knopov, Optimal′nye otsenki parametrov stokhasticheskikh sistem, “Naukova Dumka”, Kiev, 1981 (Russian). MR 619692
- A. Ya. Dorogovtsev, Teoriya otsenok parametrov sluchaĭ nykh protsessov, “Vishcha Shkola”, Kiev, 1982 (Russian). MR 668517
- N. N. Leonenko and A. V. Ivanov, Statisticheskiĭ analiz sluchaĭ nykh poleĭ, “Vishcha Shkola”, Kiev, 1986 (Russian). With a preface by A. V. Skorokhod. MR 917486
- Jan Beran, Statistics for long-memory processes, Monographs on Statistics and Applied Probability, vol. 61, Chapman and Hall, New York, 1994. MR 1304490
- V. V. Anh, V. P. Knopova, and N. N. Leonenko, Continuous-time stochastic processes with cyclical long-range dependence, Aust. N. Z. J. Stat. 46 (2004), no. 2, 275–296. MR 2076396, DOI https://doi.org/10.1111/j.1467-842X.2004.00329.x
References
- M. G. Serebrennikov and A. A. Pervozvanskiĭ, The Detection of Latent Periodicities, Nauka, Moscow, 1965. (Russian)
- P. Whittle, The simultaneous estimation of a time series harmonic components and covariance structure, Trabajos Estadistica 3 (1952), 43–57. MR 0051487 (14:488a)
- A. M. Walker, On the estimation of a harmonic component in a time series with stationary dependent residuals, Adv. Appl. Probab. 5 (1973), 217–241. MR 0336943 (49:1716)
- E. J. Hannan, The estimation of frequency, J. Appl. Probab. 10 (1973), 510–519. MR 0370977 (51:7200)
- A. V. Ivanov, A solution of the problem of detecting hidden periodicities, Teor. Veroyatnost. i Mat. Statist. 20 (1979), 44–59; English transl. in Theor. Probab. Math. Statist. 20 (1980), 51–68. MR 529259 (80c:62107)
- A. V. Ivanov, Asymptotic Theory of Nonlinear Regression, Kluwer Academic Publishers, Dordrecht, 1997. MR 1472234 (99h:62086)
- P. S. Knopov, Optimal Estimators of Parameters of Stochastic Systems, Naukova Dumka, Kiev, 1981. (Russian) MR 619692 (83d:62134)
- A. Ya. Dorogovtsev, The Theory of Estimators of Random Processes, Vyshcha Shkola, Kiev, 1982. (Russian) MR 668517 (84h:62122)
- A. V. Ivanov and N. N. Leonenko, Statistical Analysis of Random Fields, Vyshcha Shkola, Kiev, 1986; English transl., Kluwer Academic Publishers, Dordrecht, 1989. MR 917486 (89e:62125)
- J. Beran, Statistics for Long-Memory Processes, Chapman and Hall, New York, 1994. MR 1304490 (96b:62138)
- V. V. Anh, N. N. Leonenko, and V. P. Knopova, Continuous-time stochastic processes with cyclical long-range dependence, Aust. N. Z. J. Stat. 46 (2004), no. 2, 275–296. MR 2076396 (2005e:60079)
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2000):
62J02,
62J99
Retrieve articles in all journals
with MSC (2000):
62J02,
62J99
Additional Information
A. V. Ivanov
Affiliation:
Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine “KPI”, Peremogy Avenue 37, Kiev 03056, Ukraine
Email:
ivanov@paligora.kiev.ua
Keywords:
The detection of hidden periodicities,
least squares estimator,
consistency,
long range dependence
Received by editor(s):
September 11, 2008
Published electronically:
August 18, 2010
Article copyright:
© Copyright 2010
American Mathematical Society
