Stochastic asymptotic expansion of correlogram estimator of the correlation function of random noise in nonlinear regression model

Authors:
O. V. Ivanov and K. K. Moskvichova

Translated by:
N. Semenov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **90** (2014).

Journal:
Theor. Probability and Math. Statist. **90** (2015), 87-101

MSC (2010):
Primary 60G50, 65B10, 60G15; Secondary 40A05

DOI:
https://doi.org/10.1090/tpms/951

Published electronically:
August 6, 2015

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Abstract | References | Similar Articles | Additional Information

Abstract: A correlogram estimator of the covariance function of a stationary Gaussian noise is considered in a nonlinear regression model with continuous time. The estimator is constructed from deviations of the observed stochastic process from the regression function where the least squares estimator is substituted for the unknown parameter. A stochastic asymptotic expansion of the correlogram estimator of the covariance function is obtained for the case where the time of observations tends to infinity.

**1.**A. V. Ivanov,*Asymptotic Theory of Nonlinear Regression*, Kluwer Academic Publishers, Dordrecht-Boston-London, 1997. MR**1472234 (99h:62086)****2.**O. V. Ivanov and I. K. Matsak,*Limit theorems for the maximal residuals in linear and nonlinear regression models*, Teor. Imovirnost. Matem. Statyst.**86**(2012), 81-91; English transl. in Theor. Probability and Math. Statist.**86**(2013), 79-91. MR**2986451****3.**A. V. Ivanov and N. N. Leonenko,*Statistical Analysis of Random Fields*, ``Vyshcha shkola'', Kiev, 1986; English transl., Kluwer Academic Publishers, Dordrecht, 1989. MR**917486 (89e:62125)****4.**V. V. Buldygin and Yu. V. Kozachenko,*Metric Characterization of Random Variables and Random Processes*, ``TViMS'', Kiev, 1998; English transl., American Mathematical Society, Providence, RI, 2000. MR**1743716 (2001g:60089)****5.**I. I. Gihman and A. V. Skorohod,*Introduction to the Theory of Random Processes*, ``Nauka'', Moscow, 1965; English transl., Saunders, Philadelphia, 1969. MR**0198534 (33:6689)**

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Additional Information

**O. V. Ivanov**

Affiliation:
Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine “Kiev Polytechnic Institute”, Peremogy Avenue, 37, Kyiv 03056, Ukraine

Email:
alexntuu@gmail.com

**K. K. Moskvichova**

Affiliation:
Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine “Kiev Polytechnic Institute”, Peremogy Avenue, 37, Kyiv 03056, Ukraine

Email:
kamok@ua.fm

DOI:
https://doi.org/10.1090/tpms/951

Keywords:
Nonlinear regression model with continuous time,
stationary Gaussian noise,
covariance function,
least squares estimator,
stochastic asymptotic expansion

Received by editor(s):
July 31, 2013

Published electronically:
August 6, 2015

Article copyright:
© Copyright 2015
American Mathematical Society