On the least squares estimator asymptotic normality of the multivariate symmetric textured surface parameters
Authors:
A. V. Ivanov and I. M. Savych
Journal:
Theor. Probability and Math. Statist. 105 (2021), 151-169
MSC (2020):
Primary 62J02; Secondary 62J99
DOI:
https://doi.org/10.1090/tpms/1161
Published electronically:
December 7, 2021
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Abstract: A multivariate trigonometric regression model is considered. Various discrete modifications of the similar bivariate model received serious attention in the literature on signal and image processing due to multiple applications in the analysis of symmetric textured surfaces. In the paper asymptotic normality of the least squares estimator for amplitudes and angular frequencies is obtained in multivariate trigonometric model assuming that the random noise is a homogeneous or homogeneous and isotropic Gaussian, in particular, strongly dependent random field on $\mathbb {R}^M,\,\, M>2.$
References
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References
- V. Anh, N. Leonenko, and A. Olenko, On the rate of convergence to Rosenblatt-type distribution, J. Math. Anal. Appl. 425 (2015), 111–132. MR 3299653
- V. Anh, N. Leonenko, A. Olenko, and V. Vaskovych, On the rate of convergence in non-central limit theorems, Bernoulli 25 (2019), no. 4A, 2920–2948. MR 4003569
- P. Billingsley, Convergence of Probability Measures, Second Edition, Wiley, 1999. MR 1700749
- D. R. Brillinger, Regression for randomly sampled spatial series: the trigonometric case, J. Appl. Probab. 23A (1986), 275–289. MR 803178
- J. M. Francos, A. Z. Meiri, and B. Porat, A united texture model based on 2-D Wald type decomposition, IEEE Transactions on Signal Processing 17 (1993), no. 41, 2665–2678.
- 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, Application of Mathematics vol. 9, Springer-Verlag, New York-Berlin, 1978. Translated from the Russian by A. B. Aries. MR 543837
- V. I. Istratescu, Fixed Point Theory, An introduction, Mathematics and its Applications vol. 7, Springer Netherlands, 1981. MR 620639
- A. V. Ivanov, Asymptotic Theory of Nonlinear Regression, vol. 389, Kluwer Academic Publishers Group, Dordrecht, 1997. MR 1472234
- 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
- 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. Skorohod; Translated from the Russian by A. I. Kochubinskií. MR 1009786
- A. V. Ivanov, N. N. Leonenko, M. D. Ruiz-Medina, and I. N. Savich, Limit theorems for weighted non-linear transformations of Gaussian processes with singular spectra, Ann. Probab. 41 (2013), no. 2, 1088–1114. MR 3077537
- A. V. Ivanov, N. N. Leonenko, M. D. Ruiz-Medina, and B. M. Zhurakovsky, Estimation of harmonic component in regression with cyclically dependent errors, Statistics 49 (2015), no. 1, 156–186. MR 3304373
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- P. S. Knopov, Optimal’nye otsenki parametrov stokhasticheskih sisrem, “Naukova Dumka”, Kiev, 1981 (Russian). MR 619692
- D. Kundu and A. Mitra, Asymptotic properties of the least squares estimates of 2-D exponential signals, Multidimens. Systems Signal Process. 7 (1996), no. 2, 135–150. MR 1388718
- D. Kundu and S. Nandi, Determination of discrete spectrum in a random field, Statist. Neerlandica 57 (2003), no. 2, 258–283. MR 2028915
- P. Malliavan, Estimation d’un signal Lorentzien, C. R. Acad. Sci. Paris Ser. I Math. 319 (1994), no. 9, 991–997. MR 1302805
- P. Malliavan, Sur la norme d’une matrice circulante gaussienne, C. R. Acad. Sci. Paris Ser. I Math. 319 (1994), no. 7, 745–749. MR 1300081
- S. Nandi, D. Kundu, and R. K. Srivastava, Noise space decomposition method for two-dimensional sinusoidal model, Comput. Statist. Data Anal. 58 (2013), 147–161. MR 2997932
- J. Pfanzagl, On the measurability and consistency of minimum contrast estimates, Metrika 14 (1969), 249–272.
- C. R. Rao, L. C. Zhao, and B. Zhou, Maximum likelihood estimation of 2-D superimposed exponential, IEEE Transactions on Signal Processing 42 (1994), 795–802.
- 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
- J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965. MR 0184422
- M. I. Yadrenko, Spectral Theory of Random Fields (Translations Series in Mathematics and Engineering), Springer, 1983. MR 697386
- 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, 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. 22 (2001), no. 5, 613–629. MR 1859568
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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 “Igor Sikorsky Kyiv Polytechnic Institute”, Peremohy Avenue, 37, Kyiv 03057, Ukraine
Email:
alexntuu@gmail.com
I. M. Savych
Affiliation:
Department of Mathematical Analysis and Probability Theory, Faculty of Physics and Mathematics, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Peremohy Avenue, 37, Kyiv 03057, Ukraine
Email:
sim7ka@gmail.com
Keywords:
Multivariate trigonometric model,
texture surface,
homogeneous and isotropic Gaussian random field,
covariance function,
spectral density,
least squares estimate in the Walker sense,
linearization theorem,
asymptotic uniqueness,
spectral measure of regression function,
Brouwer fixed-point theorem,
$\mu$-admissibility,
asymptotic normality
Received by editor(s):
July 28, 2021
Published electronically:
December 7, 2021
Article copyright:
© Copyright 2021
Taras Shevchenko National University of Kyiv