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 $\mathbb {R}^M,\,\, M>2.$

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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**
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*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**
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*Gaussian Random Processes*, Application of Mathematics vol. 9, Springer-Verlag, New York-Berlin, 1978. Translated from the Russian by A. B. Aries. MR **543837**
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*Fixed Point Theory*, An introduction, Mathematics and its Applications vol. 7, Springer Netherlands, 1981. MR **620639**
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*Asymptotic Theory of Nonlinear Regression*, vol. 389, Kluwer Academic Publishers Group, Dordrecht, 1997. MR **1472234**
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*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**
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*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**
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*Limit theorems for weighted non-linear transformations of Gaussian processes with singular spectra*, Ann. Probab. **41** (2013), no. 2, 1088–1114. MR **3077537**
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*Estimation of harmonic component in regression with cyclically dependent errors*, Statistics **49** (2015), no. 1, 156–186. MR **3304373**
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*The asymptotic normality for the least squares estimator of parameters in a two dimensional sinusoidal model of observations*, Theor. Probab. Math. Statist. **100** (2020), 107–131. MR **3992995**
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*Consistency of the least squares estimator for the parameters of a sinusoidal model of a textured surface*, Theor. Probab. Math. Statist. **97** (2018), 73–84. MR **3746000**
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*Optimal’nye otsenki parametrov stokhasticheskih sisrem*, “Naukova Dumka”, Kiev, 1981 (Russian). MR **619692**
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*Asymptotic properties of the least squares estimates of 2-D exponential signals*, Multidimens. Systems Signal Process. **7** (1996), no. 2, 135–150. MR **1388718**
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*Determination of discrete spectrum in a random field*, Statist. Neerlandica **57** (2003), no. 2, 258–283. MR **2028915**
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*Estimation d’un signal Lorentzien*, C. R. Acad. Sci. Paris Ser. I Math. **319** (1994), no. 9, 991–997. MR **1302805**
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*Sur la norme d’une matrice circulante gaussienne*, C. R. Acad. Sci. Paris Ser. I Math. **319** (1994), no. 7, 745–749. MR **1300081**
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*On the measurability and consistency of minimum contrast estimates*, Metrika **14** (1969), 249–272.
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*Maximum likelihood estimation of 2-D superimposed exponential*, IEEE Transactions on Signal Processing **42** (1994), 795–802.
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*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**
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*The algebraic eigenvalue problem*, Clarendon Press, Oxford, 1965. MR **0184422**
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*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