Consistency of the Correlogram Estimator of Covariance Function of Random Noise in Nonlinear Regression

Олександр Володимирович Іванов, Катерина Костянтинівна Москвичова

Abstract


Background. For nonlinear regression model with continuous time and mean square continuous separable measurable Gaussian stationary random noise with zero mean and square-integrable spectral density the problem of statistical  estimation of unknown covariance function of random noise in the presence of regression function nuisance parameter is considered.

Objective. To study asymptotic behavior of residual correlogram estimator of random noise covariance function using consistent least squares estimator of regression function unknown parameter.

Methods. To obtain the paper results we rely on the use of methodological machinery of monographs by V.V. Buldygin and Yu.V. Kozachenko (2000), specifically, the theory of quadratically Gaussian random processes, and A.V. Ivanov and N.N. Leonenko (1989).

Results. Sufficient conditions of weak consistency in unknown metric of residual correlogram estimator are obtained. The rate of covariance to zero of the strong consistency in unknown metric of the estimator is formulated.

Conclusions. The results obtained given opportunity to continue research of asymptotic properties of Gaussian stationary random noise covariance function residual correlogram estimator in nonlinear regression model and to prove a functional theorem in the space of continuous function on asymptotic confidence intervals for unknown covariance function of random noise.

Keywords


Nonlinear regression model; Stationary Gaussian noise; Covariance function; Residual correlogram estimator; Consistency; Pseudometric; Metric massiveness; Metric entropy

References


O.V. Ivanov and K.K. Moskvychova, “Stochastic asymptotic expansion of random noise covariance function сorrelogram estimator in nonlinear regression model”, Teoria Imovirnostey i Matematychna Statystyka, no. 90, pp. 78–91, 2014 (in Ukrainian).

O.V. Ivanov and K.K. Moskvychova, “Asymptotic expansion of the moments of random noise covariance function сorre­logram estimator in nonlinear regression model”, Ukrainsky Matematychny Zhurnal, vol. 66, no. 6, pp. 787–805, 2014 (in Ukrainian).

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

V.V. Buldygin and Yu.V. Kozachenko, Metric Characterization of Random Variables and Random Processes (Transl. of Math. Monographs). Am. Math. Soc., 2000, 257 p.

I.K. Matsak, Elements of the Theory of Extreme Values.Kyiv,Ukraine: Comprint, 2014, 210 p. (in Ukrainian).

N.N. Leonenko, Limit Theorems for Random Fields with Singular Spectrum. Dordecht, Boston, London: Kluwer Academic Publishers, 1999, 401 p.

A.V. Ivanov, Asymtotic Theory of Nonlinear Regression. Dordecht, Boston, London: Kluwer Academic Publishers, 1997, 327 p.


GOST Style Citations


  1. Іванов О.В., Москвичова К.К. Стохастичний асимптотичний розклад корелограмної оцінки коваріаційної функції випадкового шуму в нелінійній моделі регресії // Теорія ймовірностей і мат. статистика. – 2014. – Вип. 90. – С. 78–91.

  2. Іванов О.В., Москвичова К.К. Асимптотичний розклад моментів корелограмної оцінки коваріаційної функції випадкового шуму в нелінійній моделі регресії // Укр. мат. журнал. – 2014. – 66, № 6. ­– С. 787–805.

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

  4. Buldygin V.V., Kozachenko Yu.V. Metric characterization of random variables and random processes (Transl. of Math. Monographs). – Am. Math. Soc., 2000. – 257 p.

  5. Мацак І. Елементи теорії екстремальних значень. – К.: ЦП “Компринт”, 2014. – 210 с.

  6. Leonenko N.N. Limit Theorems for Random Fields with Singular Spectrum. – Dordrecht;Boston;London: Kluwer Academic Publishers, 1999. – 401 p.

  7. Ivanov A.V. Asymtotic Theory of Nonlinear Regression. – Dordrecht; Boston; London: Kluwer Academic Publishers, 1997. – 327 p.




DOI: http://dx.doi.org/10.20535/1810-0546.2015.4.50483

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