Cross-correlogram estimators of impulse response functions

Authors:
Yu. V. Kozachenko and I. V. Rozora

Translated by:
S. Kvasko

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **93** (2015).

Journal:
Theor. Probability and Math. Statist. **93** (2016), 79-91

MSC (2010):
Primary 62M20, 60E15

DOI:
https://doi.org/10.1090/tpms/995

Published electronically:
February 7, 2017

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Abstract | References | Similar Articles | Additional Information

Abstract: The integral cross-correlogram estimator of the response function for a linear homogeneous system is considered in this paper. An upper bound for the tail of the distribution of the supremum of the estimation error is found. In the proof, we use some properties of square-Gaussian stochastic processes.

**1.**I. P. Blazhievs'ka,*Asymptotic unbiasedness and consistency of correlogram estimators of impulse response functions in linear homogeneous systems*, Naukovi Visti NTUU ``KPI''**4**(2014), 7-12. (Ukrainian)**2.**V. V. Buldygin and I. P. Blazhievs'ka,*Correlation properties of correlogram estimators of impulse response functions*, Naukovi Visti NTUU ``KPI''**5**(2009), 120-128. (Ukrainian)**3.**V. V. Buldygin and I. P. Blazhievs'ka,*Asymptotic properties of correlogram estimators of impulse response functions in linear systems*, Naukovi Visti NTUU ``KPI''**4**(2010), 16-27. (Ukrainian)**4.**I. I. Gikhman and A. V. Skorokhod,*Introduction to the theory of random processes*, Translated from the Russian by Scripta Technica, Inc, W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont., 1969. MR**0247660****5.**Yu. V. Kozachenko, A. O. Pashko, and I. V. Rozora,*Modelling Stochastic Processes and Ransom Fields*, ``Zadruga'', Kyiv, 2007. (Ukrainian)**6.**V. V. Buldygin and Yu. V. Kozachenko,*Metric characterization of random variables and random processes*, Translations of Mathematical Monographs, vol. 188, American Mathematical Society, Providence, RI, 2000. Translated from the 1998 Russian original by V. Zaiats. MR**1743716****7.**Valerij Buldygin and Viktor Kurotschka,*On cross-correlogram estimators of the response function in continuous linear systems from discrete observations*, Random Oper. Stochastic Equations**7**(1999), no. 1, 71–90. MR**1677746**, https://doi.org/10.1515/rose.1999.7.1.71**8.**V. V. Buldygin and Fu Li,*On asymptotic normality of an estimation of unit impulse responses of linear system I*, Teor. Ĭmovir. Mat. Stat.**54**(1996), 16-24; Enlglish transl. in Theor. Probability and Math. Statist.**54**(1997), 3-17.**9.**V. V. Buldygin and Fu Li,*On asymptotic normality of an estimation of unit impulse responses of linear system II*, Teor. Ĭmovir. Mat. Stat.**55**(1996), 30-37; English transl. in Theor. Probability and Math. Statist.**55**(1997), 30-37.**10.**Valery Buldygin, Frederic Utzet, and Vladimir Zaiats,*Asymptotic normality of cross-correlogram estimates of the response function*, Stat. Inference Stoch. Process.**7**(2004), no. 1, 1–34. MR**2041907**, https://doi.org/10.1023/B:SISP.0000016454.89610.35**11.**V. Buldygin, F. Utzet, and V. Zaiats,*A note on the application of integrals involving cyclic products of kernels*, Qüestiió (2)**26**(2002), no. 1-2, 3–14. MR**1924680****12.**Yurij V. Kozachenko and Olexander V. Stus,*Square-Gaussian random processes and estimators of covariance functions*, Math. Commun.**3**(1998), no. 1, 83–94 (English, with English and Croatian summaries). MR**1648867**

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Additional Information

**Yu. V. Kozachenko**

Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine

Email:
yvk@univ.kiev.ua

**I. V. Rozora**

Affiliation:
Department of Applied Statistics, Faculty for Cybernetics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine

Email:
irozora@bigmir.net

DOI:
https://doi.org/10.1090/tpms/995

Keywords:
Correlogram,
impulse response function,
large deviation probabilities

Received by editor(s):
July 9, 2015

Published electronically:
February 7, 2017

Additional Notes:
The paper was prepared following the talk at the International Conference “Probability, Reliability and Stochastic Optimization (PRESTO-2015)” held in Kyiv, Ukraine, April 7–10, 2015

Article copyright:
© Copyright 2017
American Mathematical Society