Asymptotic Uniqueness of the Non-Linear Regression Model Parameter with Least Squares Estimator

Богдан Михайлович Жураковський


In the paper the nonlinear regression model with continuous time and random noise, which is a local functional of strongly dependent stationary Gaussian random process, is considered. Sufficient conditions of asymptotic uniqueness of the least squares estimator of regression function parameters are obtained. This result is applied to the least squares estimator of amplitude and angular frequencies of harmonic oscillations sum observed on the background of given random noise. To obtain the main result limit theorems of random processes, weak convergence of a family of measures to the spectral measure of a regression function, etc were used. The novelty, compared with the known results in the theory of periodogram estimator in observation models on weakly dependent noise, is assuming that the random noise is a local functional of Gaussian strongly dependent stationary process. The result can be used in the proof of the asymptotic normality of the least squares estimator of nonlinear regression model parameters with the help of Brower fixed point theorem.


Least square estimator; Asymptotic uniqueness; Strong dependence; Hidden periodicities; Nonlinear regression


S. Chatterjee and V.C. Vani, “An Extended Matched Filtering Methods to Detect Periodicities in a Rough Grating for Extremely Large Roughness”, Bull. Astronomical Soc.India, vol. 31, pp. 457–459, 2003.

S. Chatterjee and V.C. Vani, “Scattering of light by a periodic structure in the presence of randomness. V. Detection of successive peaks in a periodic structure”, Applied Optics, vol. 45, pp. 8939–8944, 2006.

H. Arsham, “A test sensitive to extreme hidden perio­dicities, Stochastic Environmental Research and Risk Assessment”, vol. 11, no. 4, pp. 323–330, 1997.

J. Malisic et al., “Application of some statistical tests for hidden periodicity in the Serbian annual precipitation sums”, Hungarian Meteorolog. Service, vol. 103, no. 4, pp. 237–247, 1999.

Гончаренко Ю.В., Ляшко С.И. Теорема Брауэра. – К.: КИЙ, 2000. – 48 c.

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

A.V. Ivanov and I.V. Orlovsky, “Estimates in Nonli­near Regression withLongRange Dependence”, Theory of Stochastic Processes, vol. 7 (23), no. 3-4, pp. 38–39, 2002.

Иванов О.В. Конзистентність оцінки найменших квад­ратів амплітуд та кутових суми гармонійних коливань у моделях з сильною залежністю // Теорія ймовір­ності та математичної статистики. – 2009. – Вип. 80. – C. 55–62.

A.V. Ivanov and B.M. Zhurakovskyi, “Detection of hidden periodicities in the model with long range dependent noise” in Int. Conf. “Modern Stochastic: Theory and Applications II”, Ukraine, Kyiv, 7–11 September, 2010, pp. 99–100.

GOST Style Citations





  • There are currently no refbacks.

Copyright (c) 2017 NTUU KPI