Asymptotic properties of periodogram estimators in the trigonometric model for observations on the plane
Authors:
O. V. Ivanov and O. V. Lymar
Translated by:
N. N. Semenov
Journal:
Theor. Probability and Math. Statist. 101 (2020), 129-151
MSC (2020):
Primary 62J02; Secondary 62J99
DOI:
https://doi.org/10.1090/tpms/1117
Published electronically:
January 5, 2021
Full-text PDF
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References |
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Additional Information
Abstract: The simplest sinusoidal model of a symmetric textured surface observed with a noise being a homogeneous and isotropic Gaussian random field (in particular, a strongly dependent random field) on the plane is studied in the paper. The strong consistency and asymptotic normality of periodogram estimators of the amplitude and angular frequencies are proved for this model of the trigonometric regression.
References
- J. M. Francos, A. Z. Meiri, and B. Porat, A united texture model based on 2-D Wald type decomposition, IEEE Trans. Signal Processing 17 (1993), no. 41, 2665–2678.
- Rama Chellappa and Anil Jain (eds.), Markov random fields, Academic Press, Inc., Boston, MA, 1993. Theory and application. MR 1214376
- Hao Zhang and V. Mandrekar, Estimation of hidden frequencies for 2D stationary processes, J. Time Ser. Anal. 22 (2001), no. 5, 613–629. MR 1859568, DOI https://doi.org/10.1111/1467-9892.00244
- Swagata Nandi, Debasis Kundu, and Rajesh Kumar Srivastava, Noise space decomposition method for two-dimensional sinusoidal model, Comput. Statist. Data Anal. 58 (2013), 147–161. MR 2997932, DOI https://doi.org/10.1016/j.csda.2011.03.002
- Paul Malliavin, Sur la norme d’une matrice circulante gaussienne, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 7, 745–749 (French, with English and French summaries). MR 1300081
- Paul Malliavin, Estimation d’un signal lorentzien, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 9, 991–997 (French, with English and French summaries). MR 1302805
- O. V. Īvanov and O. V. Malyar, Consistency of the least squares estimator for the parameters of a sinusoidal model of a textured surface, Teor. Ĭmovīr. Mat. Stat. 97 (2017), 72–82 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 97 (2018), 73–84. MR 3746000, DOI https://doi.org/10.1090/tpms/1049
- O. V. Īvanov and O. V. Limar, Asymptotic normality of a least squares estimator for the parameters of a two-dimensional sinusoidal observation model, Teor. Ĭmovīr. Mat. Stat. 100 (2019), 102–122 (Ukrainian, with English, Russian and Ukrainian summaries). MR 3992995
- M. Ĭ. Jadrenko, Spektral′naya teoriya sluchaĭ nykh poleĭ, “Vishcha Shkola”, Kiev, 1980 (Russian). MR 590889
- A. V. Ivanov and N. N. Leonenko, Statistical analysis of random fields, Mathematics and its Applications (Soviet Series), vol. 28, Kluwer Academic Publishers Group, Dordrecht, 1989. With a preface by A. V. Skorokhod; Translated from the Russian by A. I. Kochubinskiĭ. MR 1009786
- G. P. Grečka and A. Ja. Dorogovcev, The asymptotic properties of the periodogram estimate of the frequency and amplitude of a harmonic oscillation, Vyčisl. Prikl. Mat. (Kiev) Vyp. 28 (1976), 18–31, 150 (Russian, with English summary). MR 0451587
- B. M. Zhurakovskyi and A. V. Ivanov, Periodogram estimator properties of the parameters of the regression model with strongly dependent noise, Research Bull. NTUU “KPI” (2012), no. 4, 59–65. (Ukrainian)
- P. S. Knopov and G. D. Bila, Periodogram estimates in nonlinear regression models with long-range dependent noise, Cybernet. Systems Anal. 49 (2013), no. 4, 624–631. Translation of Kibernet. Sistem. Anal. 2013, no. 4, 163–172. MR 3231469, DOI https://doi.org/10.1007/s10559-013-9549-5
- Alexander V. Ivanov, Nikolai Leonenko, María D. Ruiz-Medina, and Irina N. Savich, Limit theorems for weighted nonlinear transformations of Gaussian stationary processes with singular spectra, Ann. Probab. 41 (2013), no. 2, 1088–1114. MR 3077537, DOI https://doi.org/10.1214/12-AOP775
- Ulf Grenander, On the estimation of regression coefficients in the case of an autocorrelated disturbance, Ann. Math. Statistics 25 (1954), 252–272. MR 62402, DOI https://doi.org/10.1214/aoms/1177728784
- 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
- T. Alodat and A. Olenko, Weak convergence of weighted additive functionals of long-range dependent fields, Teor. Ĭmovīr. Mat. Stat. 97 (2017), 9–23 (English, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 97 (2018), 1–16. MR 3745995, DOI https://doi.org/10.1090/tpms/1044
- Vo Anh, Nikolai Leonenko, Andriy Olenko, and Volodymyr Vaskovych, On rate of convergence in non-central limit theorems, Bernoulli 25 (2019), no. 4A, 2920–2948. MR 4003569, DOI https://doi.org/10.3150/18-BEJ1075
- Nikolai Leonenko and Andriy Olenko, Tauberian and Abelian theorems for long-range dependent random fields, Methodol. Comput. Appl. Probab. 15 (2013), no. 4, 715–742. MR 3117624, DOI https://doi.org/10.1007/s11009-012-9276-9
References
- J. M. Francos, A. Z. Meiri, and B. Porat, A united texture model based on 2-D Wald type decomposition, IEEE Trans. Signal Processing 17 (1993), no. 41, 2665–2678.
- T. Yuan and T. Subba Rao, Spectrum estimation for random fields with application to Markov modelling and texture classification, Markov Random Fields, Theory and Applications (R. Chellappa and A. K. Jain, eds.), Academic Press, New York, 1993. MR 1214376
- H. Zhang and V. Mandrekar, Estimation of hidden frequencies for 2D stationary processes, J. Time Ser. Anal. (2001), no. 22, 613–629. MR 1859568
- S. Nandi, D. Kundu, and R. K. Srivastava, Noise space decomposition method for two-dimensional sinusoidal model, Comp. Statist. Data Anal. 58 (2013), 147–161. MR 2997932
- P. Malliavan, Sur la norte d’une matrice circulante Gaussienne, Comptes Rendus de l’Academie des Sciences, Serie 1 (Mathematique) (1994), 45–49. MR 1300081
- P. Malliavan, Estimation d’un signal Lorentzien, Comptes Rendus de l’Academie des Sciences, Serie 1 (Mathematique) (1994), 991–997. MR 1302805
- A. V. Ivanov and O. V. Maliar, Consistency of the least squares estimator of the textured surface sinusoidal model parameters, Teor. Imovirnost. Matem. Statyst. 97 (2017), 72–82; English transl. in Theor. Probability Math. Statist. 97 (2018), 73–84. MR 3746000
- A. V. Ivanov and O. V. Lymar, Asymptotic normality of the least squares estimator of two-dimensional sinusoidal observation model parameters, Teor. Imovirnost. Matem. Statyst. 97 (2017), 72–82; English transl. in Theor. Probability Math. Statist. 100 (2020). MR 3992995
- M. I. Yadrenko, Spectral Theory of Random Fields, “Vyshcha Shkola”, Kiev, 1980; English transl. Optimization Software, New York, 1983. MR 590889
- A. V. Ivanov and N. N. Leonenko, Statistical Analysis of Random Fields, Kluwer Academic Publishers, Dordecht–Boston–London, 1989. MR 1009786
- G. P. Hrechka and A. Ya. Dorogovtsev, On asymptotic properties of periodogram estimator of harmonic oscillation frequency and amplitude, Vychisl. Prikl. Mat. 28 (1975), 18–31. (Russian) MR 0451587
- B. M. Zhurakovskyi and A. V. Ivanov, Periodogram estimator properties of the parameters of the regression model with strongly dependent noise, Research Bull. NTUU “KPI” (2012), no. 4, 59–65. (Ukrainian)
- P. S. Knopov and G. D. Bila, Periodogram estimates in the nonlinear regression models with strongly dependent noise, Kibernetika Sistem. Anal. (2013), no. 4, 163–172; English transl. in Cybernetics Systems Anal. (2013), vol. 49, no. 4, 624–631 MR 3231469
- A. V. Ivanov, N. Leonenko, M. D. Ruiz-Medina, and I. N. Savich, Limit theorems for weighted nonlinear transformations of Gaussian stationary processes with singular spectra, Ann. Probab. 41 (2013), no. 2, 1088–1114. MR 3077537
- U. Grenander, On the estimation of regression coefficients in the case of an autocorrelated disturbance, Ann. Math. Statist. 25 (1954), no. 2, 252–272. MR 62402
- I. A. Ibragimov and Y. A. Rozanov, Gaussian Random Processes, “Nauka”, Moscow, 1970; English transl. Springer-Verlag, New York, 1978. MR 543837
- T. Alodat and A. Olenko, Weak convergence of weighted additive functionals of long-range dependent fields, Teor. Imovirnost. Matem. Statyst. 97 (2017), 9–23; English transl. in Theor. Probability and Math. Statist. 97 (2018), 1–16. MR 3745995
- V. Anh, N. Leonenko, A. Olenko, and V. Vaskovich, On rate of convergence in non-central limit theorems, Bernoulli 25 (2019), no. 4A, 2920–2948. MR 4003569
- N. Leonenko and A. Olenko, Tauberian and Abelian theorems for long-range dependent random fields, Comp. Appl. Probab. 15 (2013), 715–742. MR 3117624
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Additional Information
O. V. Ivanov
Affiliation:
Department of Mathematical Analysis and Probability Theory, Faculty for Physics and Mathematics, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Peremogy Avenue, 37, Kyiv 03057, Ukraine
Email:
alexntuu@gmail.com
O. V. Lymar
Affiliation:
Department of Mathematical Analysis and Probability Theory, Faculty for Physics and Mathematics, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Peremogy Avenue, 37, Kyiv 03057, Ukraine
Email:
malyar.ol95@gmail.com
Keywords:
Homogeneous and isotropic Gaussian random field,
spectral density,
amplitude,
angular frequency,
periodogram estimator,
spectral measure of a vector function,
consistency,
asymptotic normality
Received by editor(s):
August 3, 2019
Published electronically:
January 5, 2021
Article copyright:
© Copyright 2020
American Mathematical Society
