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On Statistical Properties of the Estimator of Impulse Response Function

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Stochastic Processes, Statistical Methods, and Engineering Mathematics (SPAS 2019)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 408))

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Abstract

In this paper a time-invariant continuous linear system is considered with a real-valued impulse response function which is defined on bounded domain. A sample input-output cross-correlogram is taken as an estimator of the response function. The input processes are supposed to be zero-mean stationary Gaussian processes that can be represented as the truncated series of Fourier expansion. A criterion on the shape of the impulse response function is given. For this purpose, a theory of square-Gaussian stochastic processes is used.

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References

  1. Abramovich, F., Pensky, M., Rozenholc, Y.: Laplace deconvolution with noisy observations. Electron. J. Stat. 7, 1094–1128 (2013)

    Article  MATH  Google Scholar 

  2. Akaike, H.: On the statistical estimation of the frequence response function of a system having multiple input. Ann. Inst. Stat. Math. 17, 185–210 (1965)

    Article  MATH  Google Scholar 

  3. Akaike, H.: On the use of non-Gaussian process in the identification of a linear dynamic system. Ann. Inst. Stat. Math. 18, 269–276 (1966)

    Article  MATH  Google Scholar 

  4. Alquier, P., Gautier, E., Stoltz, G.: Inverse Problems and High-Dimensional Estimation. Springer, New York (2011)

    Book  MATH  Google Scholar 

  5. Bendat, J.S., Piersol, A.G.: Engineering Applications of Correlation and Spectral Analysis. Wiley, New York (1980)

    MATH  Google Scholar 

  6. Blazhievska, I.P.: Correlogram estimation of response functions of linear systems in scheme of some independent samples. Theory Stoch. Proc. 17(33), 16–27 (2011)

    Google Scholar 

  7. Buldygin, V.V., Blazhievska, I.P.: Asymptotic properties of cross-correlogram estimator for impulse response functions of linear system. Naukovi Visti NTUU “KPI” 4, 16–27 (2010) (in Ukrainian)

    Google Scholar 

  8. Buldygin, V.V., Kozachenko, Yu.V.: Metric Characterization of Random Variables and Random Processes. Amer. Math. Soc, Providence, RI (2000)

    Book  MATH  Google Scholar 

  9. Buldygin, V.V., Fu, L.: On asymptotic normality of an estimation of unit impulse responses of linear system I, II. Theor. Probab. Math. Stat. 54, 3–17, 30–37 (1997)

    Google Scholar 

  10. Buldygin, V., Utzet, L., Zaiats, V.: Asymptotic normality of cross-correlogram estimators of the response function. Stat. Inference Stoch. Process. 7, 1–34 (2004)

    Article  MATH  Google Scholar 

  11. Cardot, H., Johannes, J.: Thresholding projection estimators in functional linear models. J. Multivar. Anal. 101, 395–408 (2010)

    Article  MATH  Google Scholar 

  12. Dashti, M., Stuart, A.M.: The Bayesian Approach to Inverse Problems. Springer, Handbook of Uncertainty Quantification (2017)

    Book  Google Scholar 

  13. Delaigle, A., Hall, P., Meister, A.: On deconvolution with repeated measurements. Ann. Stat. 36, 665–685 (2008)

    Article  MATH  Google Scholar 

  14. Efromovich, S., Kolchinskii, V.: On inverse problems with unknown operators. IEEE Transf. Inf. Theory 47, 2876–2894 (2001)

    Article  MATH  Google Scholar 

  15. Fan, J., Gijbels, I.: Local Polynomial Modelling and its Applications. Monographs on Statistics and Applied Probability, vol. 66. Chapman & Hall, London (1996)

    Google Scholar 

  16. Gikhman, I.I., Skorokhod, A.V.: Introduction to the Theory of Random Processes. Dover Publications Inc., Mineola, New York (1996)

    MATH  Google Scholar 

  17. Gondenshluger, A., Pereverzev, S.: On adaptive inverse estimation of linear functionals in Hilbert scales. Bernoulli 9(5), 783–807 (2003)

    Google Scholar 

  18. Hannan, E., Deistler, M.: The Statistical Theory of Linear Systems. Wiley, New York (1988)

    MATH  Google Scholar 

  19. Ingster, Y., Suslina, I.: Nonparametric Goodness-of-Fit Testing Under Gaussian Models. Springer, New York (2003)

    Book  MATH  Google Scholar 

  20. Kozachenko, Yu., Moklyachuk, O.: Large deviation probabilities for square-Gaussian stochastic processes. Extremes 2(3), 269–293 (1999)

    Article  MATH  Google Scholar 

  21. Kozachenko, Y., Moklyachuk, O.: Square-Gaussian stochastic processes. Theory Stoch. Process. 6(22), 3–4, 98–121 (2000)

    Google Scholar 

  22. Kozachenko, Yu., Pashko, A., Rozora, I.: Simulation of Stochastic Processes and Fields. Zadruga, Kyiv (2007).(in Ukrainian)

    MATH  Google Scholar 

  23. Kozachenko, Yu., Pogoriliak, O., Rozora, I., Tegza, A.: Simulation of Stochastic Processes with given Accuracy and Reliability. ISTE Press Ltd and Elsevier Ltd, London (2016)

    MATH  Google Scholar 

  24. Kozacenko, Yu., Rozora, I.: On cross-correlogram estimators of impulse response function. Theor. Probab. Math. Stat. 93, 79–91 (2016)

    Article  MATH  Google Scholar 

  25. Kozacenko, Y., Rozora, I.: A criterion for testing hypothesis about impulse response function. Stat. Optim. Inf. Comput. 4, 214–232 (2016)

    Google Scholar 

  26. Kozachenko, Y., Rozora, I., Turchyn, Y.: On an expansion of random processes in series. Random Oper. Stoch. Equ. 15(1), 15–34 (2007)

    Google Scholar 

  27. Kozachenko, Y., Rozora, I., Turchyn, Y.: Properties of some random series. Commun. Stat. Theory Methods 40(19-20), 3672–3683 (2011)

    Google Scholar 

  28. Kozachenko, Yu., Stus, O.: Square-Gaussian random processes and estimators of covariance functions. Math. Commun. 3(1), 83–94 (1998)

    MATH  Google Scholar 

  29. Kozachenko, Yu., Troshki, V.: A criterion for testing hypotheses about the covariance function of a stationary Gaussian stochastic process. Modern Stoch. Theory Appl. 1(1), 139–149 (2014)

    MATH  Google Scholar 

  30. Marteau, C., Sapatinas, T.: A unified treatment for non-asymptotic and asymptotic approaches to minimax signal detection. Stat. Surv. 9, 253–297 (2015)

    Google Scholar 

  31. Meister, A.: Deconvolution Problems in Nonparametric Statistics (Lecture Notes in Statistics). Springer-Verlag, Berlin (2009)

    MATH  Google Scholar 

  32. Peng, S.-L., Chen, C.-M., Huang, C.-Y., Shih, C.-T., Huang, C.-W., Chiu, S.-C., Shen, W.-C.: Effects of hemodynamic response function selection on rat fMRI statistical analyses. Front. Neurosci. 13(400), 1–7 (2019). https://doi.org/10.3389/fnins.2019.00400

  33. Pillonetto, G., Dinuzzo, F., Chen, T., de Nicolao, G., Ljung, L.: Kernel methods in system identification, machine learning and function estimation: a survey. Automatica 50(3), 657–682 (2014)

    Article  MATH  Google Scholar 

  34. Rozora, I.: Statistical hypothesis testing for the shape of impulse response function. Commun. Stat. Theory Methods 47(6), 1459–1474 (2018)

    Article  MATH  Google Scholar 

  35. Schetzen, M.: The Volterra and Wiener Theories of Nonlinear Systems. Wiley, New York (1980)

    MATH  Google Scholar 

  36. The New Palgrave Dictionary of Economics. Macmillan Publishers Ltd, third edition, UK (2018)

    Google Scholar 

  37. Söderström, T., Stoica, P.: System Identification. Prentice-Hall, London (1989)

    MATH  Google Scholar 

  38. Tsybakov, A.: Introduction to Nonparametric Estimation. Springer Series in Statistics, Springer, New York (2009)

    Book  MATH  Google Scholar 

  39. Wahba, G.: Spline Models for Observational Data, CBMSNSF Regional Conference Series in Applied Mathematics, vol 59. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1990)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

    Yuriy Kozachenko & Iryna Rozora

Authors
  1. Yuriy Kozachenko
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  2. Iryna Rozora
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Corresponding author

Correspondence to Yuriy Kozachenko .

Editor information

Editors and Affiliations

  1. Division of Mathematics and Physics, Mälardalen University, Västerås, Sweden

    Anatoliy Malyarenko

  2. Division of Mathematics and Physics, Mälardalen University, Västerås, Sweden

    Ying Ni

  3. Division of Mathematics and Physics, Mälardalen University, Västerås, Sweden

    Milica Rančić

  4. Division of Mathematics and Physics, Mälardalen University, Västerås, Sweden

    Sergei Silvestrov

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Kozachenko, Y., Rozora, I. (2022). On Statistical Properties of the Estimator of Impulse Response Function. In: Malyarenko, A., Ni, Y., Rančić, M., Silvestrov, S. (eds) Stochastic Processes, Statistical Methods, and Engineering Mathematics . SPAS 2019. Springer Proceedings in Mathematics & Statistics, vol 408. Springer, Cham. https://doi.org/10.1007/978-3-031-17820-7_25

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