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Robust Parametric Identification for ARMAX Models with Non-Gaussian and Coloured Noise: A Survey

Author

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  • Jesica Escobar

    (Instituto Politecnico Nacional ESIME Zacatenco, Unidad Profesional Adolfo Lopez Mateos, Av. IPN S/N, Mexico City 07738, Mexico
    These authors contributed equally to this work.)

  • Alexander Poznyak

    (Department of Automatic Control, CINVESTAV-IPN A.P. 14-740, Mexico City 07000, Mexico
    These authors contributed equally to this work.)

Abstract

In this paper the Cramer-Rao information bound for ARMAX (Auto-Regression-Moving-Average-Models-with-Exogenuos-inputs) under non-Gaussian noise is derived. It is shown that the direct application of the Least Squares Method (LSM) leads to incorrect (shifted) parameter estimates. This inconsistency can be corrected by the implementation of the parallel usage of the MLMW (Maximum Likelihood Method with Whitening) procedure, applied to all measurable variables of the model, and a nonlinear residual transformation using the information on the distribution density of a non-Gaussian noise, participating in Moving Average structure. The design of the corresponding parameter-estimator, realizing the suggested MLMW-procedure is discussed in details. It is shown that this method is asymptotically optimal, that is, reaches this information bound. If the noise distribution belongs to some given class, then the Huber approach (min-max version of MLM) may be effectively applied. A numerical example illustrates the suggested approach.

Suggested Citation

  • Jesica Escobar & Alexander Poznyak, 2022. "Robust Parametric Identification for ARMAX Models with Non-Gaussian and Coloured Noise: A Survey," Mathematics, MDPI, vol. 10(8), pages 1-38, April.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:8:p:1291-:d:792930
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    References listed on IDEAS

    as
    1. J. Escobar & A. Poznyak, 2011. "Time-varying matrix estimation in stochastic continuous-time models under coloured noise using LSM with forgetting factor," International Journal of Systems Science, Taylor & Francis Journals, vol. 42(12), pages 2009-2020.
    2. Virtue U. Ekhosuehi & David E. Omoregie, 2021. "Inspecting debt servicing mechanism in Nigeria using ARMAX model of the Koyck-kind," Operations Research and Decisions, Wroclaw University of Science Technology, Faculty of Management, vol. 31, pages 5-20.
    3. Virtue U. Ekhosuehi & David E. Omoregie, 2021. "Inspecting debt servicing mechanism in Nigeria using ARMAX model of the Koyck-kind," Operations Research and Decisions, Wroclaw University of Science and Technology, Faculty of Management, vol. 31(1), pages 5-20.
    4. Hickey, Emily & Loomis, David G. & Mohammadi, Hassan, 2012. "Forecasting hourly electricity prices using ARMAX–GARCH models: An application to MISO hubs," Energy Economics, Elsevier, vol. 34(1), pages 307-315.
    5. Bowden,Roger J. & Turkington,Darrell A., 1990. "Instrumental Variables," Cambridge Books, Cambridge University Press, number 9780521385824.
    6. Zadrozny, Peter, 1988. "Gaussian Likelihood of Continuous-Time ARMAX Models When Data Are Stocks and Flows at Different Frequencies," Econometric Theory, Cambridge University Press, vol. 4(1), pages 108-124, April.
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