Cross-correlogram estimators of impulse response functions
Authors:
Yu. V. Kozachenko and I. V. Rozora
Translated by:
S. Kvasko
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
MathSciNet review:
3553441
Full-text PDF Free Access
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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.
References
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- 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)
- 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)
- I. I. Gikhman and A. V. Skorokhod, Introduction to the theory of random processes, W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont., 1969. Translated from the Russian by Scripta Technica, Inc. MR 0247660
- Yu. V. Kozachenko, A. O. Pashko, and I. V. Rozora, Modelling Stochastic Processes and Ransom Fields, “Zadruga”, Kyiv, 2007. (Ukrainian)
- 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, DOI 10.1090/mmono/188
- 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, DOI 10.1515/rose.1999.7.1.71
- 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.
- 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.
- 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, DOI 10.1023/B:SISP.0000016454.89610.35
- 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
- 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
References
- 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)
- 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)
- 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)
- I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes, “Nauka”, Moscow, 1977; English transl., Scripta Technica, Inc. W. B. Saunders Co., Philadelphia, Pa.–London–Toronto, Ont. 1969 MR 0247660
- Yu. V. Kozachenko, A. O. Pashko, and I. V. Rozora, Modelling Stochastic Processes and Ransom Fields, “Zadruga”, Kyiv, 2007. (Ukrainian)
- 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
- V. V. Buldygin and V. G. Kurotschka, On cross-correlogram estimators of the response function in continuous linear systems from discrete observations, Random Oper. Stoch. Equ. 7 (1999), no. 1, 71–90. MR 1677746
- 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.
- 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.
- V. Buldygin, F. Utzet, and V. Zaiats, Asymptotic normality of cross-correlogram estimators of the response function, Stat. Inference Stoch. Process. 7 (2004), 1–34. MR 2041907
- V. Buldygin, F. Utzet, and V. Zaiats, A note on the application of intergals involving cyclic products of kernels, Qüestiió 26, no. 1–2 (2002), 3–14. MR 1924680
- Yu. V. Kozachenko and O. V. Stus, Square-Gaussian random processes and estimators of covariance functions, Math. Communications 3 (1998), no. 1, 83–94. 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
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
