Consistency of the Least Squares Estimate of Textured Surface Parameters

Alexander V. Ivanov, Oleksandra V. Maliar


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.


Textured surface; Sinusoidal model; Homogeneous and isotropic random field; Covariance function; Slowly varying function; Least squares estimate; Consistency


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