Moments Asymptotic Expansion of the Least Squares Estimator of the Vector-Parameter of Nonlinear Regression with Correlated Observations

A nonlinear regression model with continuous time and mean square continuous separable measurable Gaussian stationary random noise with zero mean and integrable covariance function is considered. Parameter estimation in the models of such kind is an important problem of statistics of random processes. In this paper, the first terms of asymptotic expansions of the bias vector and covariance matrix of the least square estimator of nonlinear regression function vector parameter are obtained. The machinery of the theory of stochastic processes and asymptotic theory of nonlinear regression were used to derive the results. In particular, the theorems on stochastic expansion of the least square estimator for smooth regression function and on strengthened consistency of the least squares estimator of the nonlinear regression model multidimensional parameter have been used. Obtained results allow answering question important in applications about asymptotic behavior of the first and second moments of the least squares estimator of nonlinear regression model parameter.

Publication year: 
2014
Issue: 
4
УДК: 
519.21
С. 67–74., Бібліогр.: 9 назв.
References: 

1. A.V. Ivanov and N.N. Leonenko, Statistical Analysis of Random Fields. Dordecht, Boston, London: Kluwer Academic Publishers, 1989, 244 p.
2. N.N. Leonenko, Limit Theorems for Random Fields with Singular Spectrum. Dordecht, Boston, London: Kluwer Academic Publishers, 1999, 401 p.
3. A.V. Ivanov and N.N. Leonenko, “Robust Estimators in Non-linear Regression Models with Long-Range Dependence”, Springer Optimization and its Applications, vol. 28, pp. 193—221, 2009.
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References [transliteration]: 

1. A.V. Ivanov and N.N. Leonenko, Statistical Analysis of Random Fields. Dordecht, Boston, London: Kluwer Academic Publishers, 1989, 244 p.
2. N.N. Leonenko, Limit Theorems for Random Fields with Singular Spectrum. Dordecht, Boston, London: Kluwer Academic Publishers, 1999, 401 p.
3. A.V. Ivanov and N.N. Leonenko, “Robust Estimators in Non-linear Regression Models with Long-Range Dependence”, Springer Optimization and its Applications, vol. 28, pp. 193–221, 2009.
4. Ivanov O.V., Savych I.M. Pro asymptotychnyĭ rozpodil ot͡sinky Koenkera–Basseta parametra rehresiï z syl′no zalez͡hnym shumom // Ukr. mat. z͡hurnal. – 2011. – 63, # 8. – S. 1030–1052.
5. A.V. Ivanov et al., “Limit Theorems for weighted nonlinear transformations of Gaussian stationary processes with singular spectra”, The Annals of Probability, vol. 41, no. 2, pp. 1088–1114, 2013.
6. Ibragimov I.A., Khas'minskiĭ R.Z. Asimptoticheskai͡a teorii͡a ot͡senivanii͡a. – M.: Nauka, Glav. red. fiz.-mat. lit-ry, 1979. – 528 s.
7. A.V. Ivanov, Asymtotic Theory of Nonlinear Regression. Dordecht, Boston, London: Kluwer Academic Publishers, 1997, 327 p.
8. G.P.Y. Clarke, “Moments of the Least Squares Estimators in a Nonlinear Regression Model”, J. Roy. Statist. Soc. B, vol. 42, pp. 227–237, 1980.
9. Gikhman I.I., Skorokhod A.V. Vvedenie v teorii͡u sluchaĭnykh prot͡sessov. – M.: Nauka, 1965. – 656 s.

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