Consistency of the least squares estimators of parameters in the texture surface sinusoidal model
Authors:
A. V. Ivanov and O. V. Malyar
Translated by:
S. V. Kvasko
Journal:
Theor. Probability and Math. Statist. 97 (2018), 73-84
MSC (2010):
Primary 62J02; Secondary 62J99
DOI:
https://doi.org/10.1090/tpms/1049
Published electronically:
February 21, 2019
MathSciNet review:
3746000
Full-text PDF
Abstract |
References |
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Additional Information
Abstract: We consider the texture surface sinusoidal model of observations. In other words, we consider a model where the regression function is the sum of two-parameter harmonic oscillations while the noise is an isotropic and homogeneous Gaussian random field on the plane. Conditions for the joint consistency of the least squares estimator of unknown amplitudes and angular frequencies are obtained for this trigonometric regression model.
References
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References
- J. M. Francos, A. Z. Meiri, and B. Porat, A united texture model based on 2-D Wald type decomposition, IEEE Trans. Signal Process. 41 (1993), 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.
- H. Zhang and V. Mandrekar, Estimation of hidden frequencies for 2D stationary processes, J. Time Series Anal. 22 (2001), 613–629. MR 1859568
- 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
- P. Malliavan, Sur la norte d’une matrice circulante Gaussienne, Comptes Rendus de l’Academie des Sciences, Serie 1 (Mathematique) (1994), 45–49.
- P. Malliavin, Estimation d’un signal Lorentzien, C. R. Acad. Sci. Paris Ser. I Math. 319 (1994), no. 9, 991–997. MR 1302805
- M. I. Yadrenko, Spectral Theory of Random Fields, Optimization Software, New York, 1983. MR 697386
- A. V. Ivanov and N. N. Leonenko, Statistical Analysis of Random Fields, Kluwer Academic Publishers, Dordecht–Boston–London, 1989. MR 1009786
- 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, Theory Probab. Math. Statist. 80 (2010), 61–69. MR 2541952
- 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), 156–186. MR 3304373
- C. R. Rao, L. C. Zhao, and B. Zhou, Maximum likelihood estimation of 2-D superimposed exponential, IEEE Trans. Signal Process. 42 (1994), 795–802.
- D. Kundu and A. Mitra, Asymptotic properties of the least squares estimates of 2-D exponential signals, Multidimens. Syst. Signal Process. 7 (1996), 135–150. MR 1388718
- D. Kundu and S. Nandi, Determination of discrete spectrum in a random field, Statistica Neerlandica 57 (2003), no. 2, 258–284. MR 2028915
- D. R. Brillinger, Regression for randomly sampled spatial series: The trigonometric case, J. Appl. Probab. 23 (1986), 275–289. MR 803178
- A. V. Ivanov, Asymptotic Theory of Nonlinear Regression, Kluwer Academic Publishers, Dordecht–Boston–London, 1997. MR 1472234
- A. M. Walker, On the estimation of a harmonic component in a time series with stationary dependent residuals, Adv. Appl. Probab. 5 (1973), 217–241. MR 0336943
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Additional Information
A. 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. Malyar
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:
malyar95@ukr.net
Keywords:
Texture surface sinusoidal model of observations,
isotropic and homogeneous random field,
least squares estimator,
consistency
Received by editor(s):
October 30, 2017
Published electronically:
February 21, 2019
Article copyright:
© Copyright 2019
American Mathematical Society
