Consistency and asymptotic normality of the periodogram estimator of harmonic oscillation parameters
Authors:
A. V. Ivanov and B. M. Zhurakovskyi
Journal:
Theor. Probability and Math. Statist. 89 (2014), 33-43
MSC (2010):
Primary 62J02; Secondary 62J99
DOI:
https://doi.org/10.1090/S0094-9000-2015-00933-9
Published electronically:
January 26, 2015
MathSciNet review:
3235173
Full-text PDF Free Access
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Abstract: The problem of detection of hidden periodicities is considered in the paper. We study the model of the harmonic oscillation observed on the background of random noise being a local functional of a Gaussian strongly dependent stationary process. To estimate unknown angular frequency and amplitude of the harmonic oscillation, the periodogram estimator is chosen. Sufficient conditions of the asymptotic normality are found for the periodogram estimator and the limit normal distribution is determined.
References
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- 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
- G. P. Grečka and A. Ya. Dorogovcev, On asymptotical properties of periodogram estimator of harmonic oscillation frequency and amplitude, Comput. Appl. Math. 28 (1976), 18–31.
- A. V. Ivanov, A solution of the problem of detecting hidden periodicities, Theor. Probab. Math. Statist. 20 (1980), 51–68.
- P. S. Knopov, Optimal′nye otsenki parametrov stokhasticheskikh sistem, “Naukova Dumka”, Kiev, 1981 (Russian). MR 619692
- B. G. Quinn and E. J. Hannan, The estimation and tracking of frequency, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 9, Cambridge University Press, Cambridge, 2001. MR 1813156
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- A. V. Ivanov and B. M. Zhurakovskyi, The least squares estimator consistency of parameters of a sum of harmonic oscillations in the models with strongly dependent noise, Naukovi visti NTUU “KPI” 4 (2010), 60–66. (Ukrainian)
- Il′dar Abdullovich Ibragimov and Y. A. Rozanov, Gaussian random processes, Applications of Mathematics, vol. 9, Springer-Verlag, New York-Berlin, 1978. Translated from the Russian by A. B. Aries. MR 543837
- A. V. Ivanov and B. M. Zhurakovskyi, Detection of hidden periodicities in the model with long range dependent noise, International Conference Modern Stochastic: Theory and Applications II, Kiev, 2010, pp. 99–100.
- O. V. Īvanov, Consistency of the least squares estimator of the amplitudes and angular frequencies of the sum of harmonic oscillations in models with strong dependence, Teor. Ĭmovīr. Mat. Stat. 80 (2009), 55–62 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 80 (2010), 61–69. MR 2541952, DOI https://doi.org/10.1090/S0094-9000-2010-00794-0
References
- M. G. Serebrennikov and A. A. Pervozvanskyi, The Detection of Hidden 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)
- G. P. Grečka and A. Ya. Dorogovcev, On asymptotical properties of periodogram estimator of harmonic oscillation frequency and amplitude, Comput. Appl. Math. 28 (1976), 18–31.
- A. V. Ivanov, A solution of the problem of detecting hidden periodicities, Theor. Probab. Math. Statist. 20 (1980), 51–68.
- P. S. Knopov, Optimal Estimators of Parameters of Stochastic Systems, “Naukova Dumka”, Kiev, 1981. (Russian) MR 619692 (83d:62134)
- B. G. Quinn and E. J. Hannan, The Estimation and Tracking of Frequency, Cambridge University Press, New York, 2001. MR 1813156 (2002g:62141)
- M. Artis, M. Hoffmann, D. Nachane, and J. Toro, The Detection of Hidden Periodicities: a Comparison of Alternative Methods, EUI Working Paper, No. ECO 2004/10, Badia Fiesolana, San Domenico (FI).
- A. V. Ivanov and B. M. Zhurakovskyi, The least squares estimator consistency of parameters of a sum of harmonic oscillations in the models with strongly dependent noise, Naukovi visti NTUU “KPI” 4 (2010), 60–66. (Ukrainian)
- I. A. Ibragimov and Y. A. Rozanov, Gaussian Random Processes, Applications of Mathematics, vol. 9, Springer-Verlag, Berlin–Heidelberg–New York, 1978. MR 543837 (80f:60038)
- A. V. Ivanov and B. M. Zhurakovskyi, Detection of hidden periodicities in the model with long range dependent noise, International Conference Modern Stochastic: Theory and Applications II, Kiev, 2010, pp. 99–100.
- A. V. Ivanov, Consistency of the least squares estimator of the amplitudes and angular frequencies of a sum of harmonic oscillations in models with long-range dependence, Theor. Probab. Math. Statist. 80 (2010), 61–69. MR 2541952 (2010f:62244)
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Additional Information
A. V. Ivanov
Affiliation:
Department of Mathematical Analysis and Probability Theory, Faculty of Physics and Mathematics, National Technical University of Ukraine, “Kiev Politechnic Institute”, Peremohy ave., 37, Kyiv 03056, Ukraine
Email:
alexntuu@gmail.com
B. M. Zhurakovskyi
Affiliation:
Department of Mathematical Analysis and Probability Theory, Faculty of Physics and Mathematics, National Technical University of Ukraine, “Kiev Politechnic Institute”, Peremohy ave., 37, Kyiv 03056, Ukraine
Email:
zhurak@gmail.com
Keywords:
Hidden periodicities,
periodogram estimator,
harmonic oscillation
Received by editor(s):
December 22, 2012
Published electronically:
January 26, 2015
Article copyright:
© Copyright 2015
American Mathematical Society
