The asymptotic normality for the least squares estimator of parameters in a two dimensional sinusoidal model of observations

Ivanov, O.; Lymar, O. V.

doi:10.1090/tpms/1100

The asymptotic normality for the least squares estimator of parameters in a two dimensional sinusoidal model of observations

Authors: O. V. Ivanov and O. V. Lymar
Translated by: S. V. Kvasko
Journal: Theor. Probability and Math. Statist. 100 (2020), 107-131
MSC (2010): Primary 62J02; Secondary 62J99
DOI: https://doi.org/10.1090/tpms/1100
Published electronically: August 4, 2020
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A two dimensional trigonometric model of observations is considered. Various discrete modifications of this model have received considerable attention in the literature on signal processing due to important applications of such models in the analysis of the textured surfaces. The asymptotic normality of the least squares estimator for amplitudes and angular frequencies is proved for this trigonometric regression model under the assumption that the random noise is a homogeneous and isotropic Gaussian, in particular, strongly dependent, random field on the plane.

References

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, and Andriy Olenko, On the rate of convergence to Rosenblatt-type distribution, J. Math. Anal. Appl. 425 (2015), no. 1, 111–132. MR 3299653, DOI https://doi.org/10.1016/j.jmaa.2014.12.016
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
R. N. Bhattacharya and R. Ranga Rao, Normal approximation and asymptotic expansions, John Wiley & Sons, New York-London-Sydney, 1976. Wiley Series in Probability and Mathematical Statistics. MR 0436272
David R. Brillinger, Regression for randomly sampled spatial series: the trigonometric case, J. Appl. Probab. Special Vol. 23A (1986), 275–289. Essays in time series and allied processes. MR 803178, DOI https://doi.org/10.1017/s0021900200117139

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.

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

Alexander V. Ivanov, Asymptotic theory of nonlinear regression, Mathematics and its Applications, vol. 389, Kluwer Academic Publishers Group, Dordrecht, 1997. MR 1472234

O. V. Īvanov, Consistency of the least squares estimator of the amplitudes and angular frequencies of the sum of harmonic oscillations in models with strong dependence, Teor. Ĭmovīr. Mat. Stat. 80 (2009), 55–62 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 80 (2010), 61–69. MR 2541952, DOI https://doi.org/10.1090/S0094-9000-2010-00794-0

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

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

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, DOI https://doi.org/10.1080/02331888.2013.864656

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

P. S. Knopov, Optimal′nye otsenki parametrov stokhasticheskikh sistem, “Naukova Dumka”, Kiev, 1981 (Russian). MR 619692

Debasis Kundu and Amit 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, DOI https://doi.org/10.1007/BF01827810

Debasis Kundu and Swagata Nandi, Determination of discrete spectrum in a random field, Statist. Neerlandica 57 (2003), no. 2, 258–283. MR 2028915, DOI https://doi.org/10.1111/1467-9574.00230

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

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

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

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, DOI https://doi.org/10.2307/1426034

J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965. MR 0184422

M. Ĭ. Jadrenko, Spektral′naya teoriya sluchaĭ nykh poleĭ, “Vishcha Shkola”, Kiev, 1980 (Russian). MR 590889

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.

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