Theory of Probability and Mathematical Statistics
Asymptotic properties of periodogram parameter estimators for a trigonometric observation model on the plane
A. V. Ivanov, O. V. Lymar
Download PDF
Abstract: The simplest sinusoidal model of symmetric texture surface is considered, which is observed on the background of homogeneous and isotropic Gaussian, in particular, strongly dependent random field on the plane. Strong consistency and asymptotic normality of periodogram amplitude and angular frequencies estimators of the specified trigonometric regression model are proved.
Keywords:
Bibliography: 1. J. M. Francos, A. Z. Meiri, B. Porat, A united texture model based on 2-D Wald type decomposition, IEEE Transactions on Signal Processing, 17 (1993), no. 41, 2665-2678.
2. T. Yuan, T. Subba Rao, Spectrum estimation for random elds with application to Markov modelling and texture classi cation, Markov Random Fields, Theory and Applications (R. Chellappa, A. K. Jain, eds.), Academic Press, New York, 1993.
3. H. Zhang, V. Mandrekar, Estimation of hidden frequencies for 2D stationary processes, Journal of Time Series Analysis, (2001), no. 22, 613-629.
4. S. Nandi, D. Kundu, R. K. Srivastava, Noise space decomposition method for two-dimensional sinusoidal model, Computational Statistics and Data Analysis, 58 (2013), 147-161.
5. P. Malliavan, Sur la norte d'une matrice circulante Gaussienne, Comptes Rendus de l'Academie des Sciences, Serie 1 (Mathematique), (1994), 45-49.
6. P. Malliavan, Estimation d'un signal Lorentzien, Comptes Rendus de l'Academie des Sciences, Serie 1 (Mathematique), (1994), 991-997.
7. A. V. Ivanov, O. V. Maliar, Consistency of the least squares estimator of the textured surface sinusoidal model parameters, Theor. Probability and Math. Statist., 97 (2017), 72-82.
8. A. V. Ivanov, O. V. Lymar, Asymptotic normality of the least squares estimator of twodimensional sinusoidal observation model parameters, Theor. Probability and Math. Statist., 100 (2019), 102-122.
9. M. I. Yadrenko, Spectral Theory of Random Fields, Optimization Software, New York, 1983.
10. A. V. Ivanov, N. N. Leonenko, Statistical Analysis of Random Fields, Kluwer Academic Publishers, Dordecht, Boston, London, 1989.
11. G. P. Hrechka, A. Ya. Dorogovtsev, On asymptotic properties of periodogram estimator of harmonic oscillation frequency and amplitude, Computing and Applied Math., 28 (1975), 18-31.
12. B. M. Zhurakovskyi, A. V. Ivanov, Periodogram estimator properties of the parameters of the regression model with strongly dependent noise, Research bulletin of NTUU "KPI (2012), no. 4,59-65.
13. P. S. Knopov, G. D. Bila, Periodogram estimates in the nonlinear regression models with strongly dependent noise, Kibernetika i Sistemny Analiz, (2013), no. 4, 163-172.
14. A. V. Ivanov, N. Leonenko, M. D. Ruiz-Medina, I. N. Savich, Limit theorems for weighted nonlinear transformations of Gaussian stationary processes with singular spectra, Annals Prob., 41 (2013), no. 2, 1088-1114.
15. U. Grenander, On the estimation of regression cou cients in the case of an autocorrelated disturbance, Ann. Math. Statist., 25 (1954), no. 2, 252-272.
16. I. A. Ibragimov, Y. A. Rozanov, Gaussian Random Processes, Springer-Verlag, New York, 1978.
17. T. Alodat, A. Olenko, Weak convergence of weighted additive functionals of long-range dependent elds, Theor. Probability and Math. Statist., 97 (2017), 9-23.
18. V. Anh, N. Leonenko, A. Olenko, V. Vaskovich, On rate of convergence in non-central limit theorems, Bernoulli, 25 (2019), no. 4A, 2920-2948.
19. N. Leonenko, A. Olenko, Tauberian and Abelian theorems for long-range dependent random elds, Comp. Appl. Prob., 15 (2013), 715-742.
