DOI: https://doi.org/10.20535/1810-0546.2017.4.103155

### Consistency of the Least Squares Estimate of Textured Surface Parameters

#### Abstract

**Background.** For sinusoidal observation model of textured surface, i.e. model where regression function is a two-parameter harmonic oscillation and noise is a homogeneous isotropic Gaussian random field on the plane with zero mean and covariance function of the special kind, the problem of statistical estimation of sinusoidal model unknown amplitudes and angular frequencies is considered.

**Objective.** The aim of the paper is to study asymptotic behavior of the least squares estimate (LSE) of unknown amplitudes and angular frequencies of textured surface sinusoidal model.

**Methods.** The results were obtained on the application of methodology developed in the papers by A.V. Ivanov et al. (2009, 2015) and monograph by A.V. Ivanov, N.N. Leonenko (1989).

**Results.** Sufficient conditions on random noise covariance function that ensure strong consistency of sinusoidal model parameters LSE are obtained.

**Conclusions.**The obtained results allow extending them on the models where regression function is a sum of several two-parameter harmonic oscillations. Besides LSE consistency property will allow proving asymptotic normality of these estimates in further works.

#### Keywords

#### Full Text:

PDF (Українська)#### References

J.M. Francos *et al.*, “A united texture model based on 2-D Wald type decomposition”, *IEEE Trans. Signal Process.*, vol. 41, pp. 2665–2678, 1993.

T. Yuan and T. Subba Rao, “Spectrum estimation for random fields with application to Markov modeling and texture classification”, in *Markov Random Fields, Theory and Applications*, R. Chellappa and A.K. Jain, eds. New York: Academic Press, 1993.

H. Zhang and V. Mandrekar, “Estimation of hidden frequencies for 2D stationary processes”, *J. Time Series Anal.*, vol. 22, pp. 613–629, 2001.

S. Nandi *et al.* “Noise space decomposition method for two-dimensional sinusoidal model”, *Computation Statistics and Data Analysis*, vol. 58, pp. 147–161, 2013. doi: 10.1016/j.csda.2011.03.002

P. Malliavan, “Estimation d’un signal Lorentzien”, *Comptes Rendus de l'Academie des Sciences. Serie 1** **(Mathematique)*, vol. 319, pp. 991–997, 1994.

A.V. Ivanov and N.N. Leonenko, *Statistical Analysis of Random Fields*. Dordecht, Boston, London: Kluwer Academic Publishers, 1989, 244 p.

O.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. Probability Math. Statist.*, vol. 80, pp. 61–69, 2010. doi: 10.1090/S0094-9000-2010-00794-0

A.V. Ivanov *et al.*, “Estimation of harmonic component in regression with cyclically dependent errors”, *Statistics: J. Theor. Appl. Statist.*, vol. 49, pp. 156–186, 2015. doi: 10.1080/02331888.2013.864656

C.R. Rao *et al.*, “Maximum likelihood estimation of 2-D superimposed exponential”, *IEEE Trans. Signal Process.*, vol. 42, pp. 795–802, 1994. doi: 10.1109/78.298285

D. Kundu and A. Mitra, “Asymptotic properties of the least squares estimates of 2-D exponential signals”, *Multidimensional Systems and Signal Processing*, vol. 7, pp. 135–150, 1997.

D. Kundu and S. Nandi, “Determination of discrete spectrum in a random field”, *Statistica Neerlandica*, vol. 57, no. 2, pp. 258–284, 2003. doi: 10.1111/1467-9574.00230

I.I. Gihman and A.V. Skorokhod,* Introduction to the Theory of Random Processes*. New York: Dover Publications, 1996.

A.V. Ivanov, “A solution of the problem of detecting hidden periodicities”, *Theor. Probab. Math. Statist.*, vol. 29, pp. 51–68, 1980.

#### GOST Style Citations

Copyright (c) 2017 Igor Sikorsky Kyiv Polytechnic Institute

This work is licensed under a Creative Commons Attribution 4.0 International License.