Periodogram Estimator Properties of the Parameters of the Modulate almost Periodic Signal

The problem of detection of hidden periodicities is considered in the paper. In the capacity of useful signal model the modulated almost periodic signal is taken observed on the background of random noise being the local functional of Gaussian strongly dependent stationary process. For estimation of unknown amplitude and angular frequency of modulated signal periodogram estimators are chosen. Sufficient conditions on consistency and asymptotic normality of the estimators are obtained. The exact form of limiting normal distribution is found. To obtain the main result there were used limit theorems of random processes, weak convergence of a family of measures to the spectral measure of a regression function, etc. The novelty, compared with the known results in the theory of periodogram estimator in observation models on weakly dependent noise, is assuming that the random noise is a local functional of Gaussian strongly dependent stationary process.

Publication year: 
2013
Issue: 
4
УДК: 
519.21
С. 45—54. Бібліогр.: 16 назв.
References: 

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References [transliteration]: 

1.P. Whittle, “The simultaneous estimation of a time series harmonic components and covariance structure”, Trabajos Estadistica, vol. 3, pp. 43–57, 1952.
2.A.M. Walker, “On the estimation of a harmonic component in a time series with stationary dependent residuals”, Adv. Appl. Probability, vol. 5, pp. 217–241, 1973.
3.E.J. Hannan, “The estimation of frequency”, Ibid, vol. 10, pp. 510–519, 1973
4.A.V. Ivanov, “A solution of the problem of detecting hidden periodicities”, Theory Probability and Math. Statist., no. 20, pp. 51–68, 1980.
5.Knopov P.S. Optimal'nye ot͡senki parametrov stokhasticheskikh sistem. – K.: Nauk. dumka, 1981. – 152 s.
6.S. Chatterjee and V.C. Vani, “An Extended Matched Filtering Methods to Detect Periodicities in a Rough Grating for Extremely Large Roughness”, Bulletin of the Astronomical Soc. of India, vol. 31, pp. 457–459, 2003.
7.A.V. Levenets et al., “Estimating signal spectra with a method of determining concealed periodicities in zero crossings”, Measurement Techniques, vol. 39, no. 9, pp. 909–913, 1996.
8.S. Chatterjee and V.C. Vani, “Scattering of light by a periodic structure in the presence of randomness. V. Detection of successive peaks in a periodic structure”, Appl. Optics, vol. 45, pp. 8939 8944, 2006.
9.M. Hinich, “Detecting a hidden periodic signal when its period is unknown”, Acoustics, Speech and Signal Processing, vol. 30, is. 5, pp. 747–750, 1982.
10.I. Iavorskyj and V. Mykhajlyshyn, “Detecting hidden periodicity of time-series generated by nonlinear processes in magneto-plasma”, Proc. of 6th Int. Conf. “Mathematical methods in Electromagnetic Theory”, is. 10–13, pp. 397–400, 1996.
11.H. Arsham, “A test sensitive to extreme hidden periodicitie”s, Stochastic Environmental Research and Risk Assessment, vol. 11, no. 4, pp. 323–330, 1997.
12.J. Malisic et al., “Application of some statistical tests for hidden periodicity in the Serbian annual precipitation sums”, Hungarian Meteorological Service, vol. 103, no. 4, pp. 237–247, 1999.
13.Knopov P.S. Ot͡senivanie neizvestnykh parametrov pochti periodicheskoĭ funkt͡sii pri nalichii shuma // Kibernetika. – 1984. – # 6. – S. 83–87.
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15.Ibragimov I.A., Rozanov I͡U.A. Gaussovy prot͡sessy. – M.: Nauka, 1971.
16.A.V. Ivanov and B.M. Zhurakovskyi, “Detection of hidden periodicities in the model with long range dependent noise”, Proc. of Int. Conf. “Modern Stochastic: Theory and Applications II”, Ukraine, Kyiv, 7–11 Spt., 2010, pp. 99–100.

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