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On Random Extended Intervals and their ARMA Processes

Author

Listed:
  • Babel Raïssa Guemdjo Kamdem

    (IMSP - Institut de Mathématiques et de Sciences Physiques - UAC - Université d’Abomey-Calavi = University of Abomey Calavi)

  • Jules Sadefo-Kamdem

    (MRE - Montpellier Recherche en Economie - UM - Université de Montpellier)

  • Carlos Ougouyandjou

    (IMSP - Institut de Mathématiques et de Sciences Physiques - UAC - Université d’Abomey-Calavi = University of Abomey Calavi)

Abstract

This work introduces and characterizes the so called "random extended intervals", these are random intervals for which the left bound may be higher than the right one. To carry out this study, we introduce on the set of random extended intervals a structure of metric space relevant to study extended interval-valued ARMA time series. This is done by extending the Hausdorff metric on extended intervals and defining a family of metrics dγ relevant for the set of random extended intervals and which do not have some disadvantages of the Hausdorff metric. We show that there exists a unique metric dγ for which γ(t)dt is what we have called "adapted measure" and we use this metric to define variability for random extended intervals.

Suggested Citation

  • Babel Raïssa Guemdjo Kamdem & Jules Sadefo-Kamdem & Carlos Ougouyandjou, 2020. "On Random Extended Intervals and their ARMA Processes," Working Papers hal-03169516, HAL.
    • Handle: RePEc:hal:wpaper:hal-03169516
    Note: View the original document on HAL open archive server: https://hal.umontpellier.fr/hal-03169516
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    References listed on IDEAS

    as
    1. Jules Sadefo Kamdem & Babel Raïssa Guemdjo Kamdem & Carlos Ougouyandjou, 2021. "S-ARMA Model and Wold Decomposition for Covariance Stationary Interval-Valued Time Series Processes," New Mathematics and Natural Computation (NMNC), World Scientific Publishing Co. Pte. Ltd., vol. 17(01), pages 191-213, March.
    2. Mbairadjim Moussa, A. & Sadefo Kamdem, J. & Terraza, M., 2014. "Fuzzy value-at-risk and expected shortfall for portfolios with heavy-tailed returns," Economic Modelling, Elsevier, vol. 39(C), pages 247-256.
    3. Wang, Xun & Zhang, Zhongzhan & Li, Shoumei, 2016. "Set-valued and interval-valued stationary time series," Journal of Multivariate Analysis, Elsevier, vol. 145(C), pages 208-223.
    4. Billard L. & Diday E., 2003. "From the Statistics of Data to the Statistics of Knowledge: Symbolic Data Analysis," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 470-487, January.
    5. Gloria González-Rivera & Wei Lin, 2013. "Constrained Regression for Interval-Valued Data," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 31(4), pages 473-490, October.
    6. Sadefo Kamdem, J. & Mbairadjim Moussa, A. & Terraza, M., 2012. "Fuzzy risk adjusted performance measures: Application to hedge funds," Insurance: Mathematics and Economics, Elsevier, vol. 51(3), pages 702-712.
    7. Mbairadjim Moussa, A. & Sadefo Kamdem, J. & Shapiro, A.F. & Terraza, M., 2014. "CAPM with fuzzy returns and hypothesis testing," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 40-57.
    8. Sun, Yuying & Han, Ai & Hong, Yongmiao & Wang, Shouyang, 2018. "Threshold autoregressive models for interval-valued time series data," Journal of Econometrics, Elsevier, vol. 206(2), pages 414-446.
    9. Ai Han & Yongmiao Hong & Shouyang Wang & Xin Yun, 2016. "A Vector Autoregressive Moving Average Model for Interval-Valued Time Series Data," Advances in Econometrics, in: Essays in Honor of Aman Ullah, volume 36, pages 417-460, Emerald Group Publishing Limited.
    10. Jules Sadefo Kamdem & Babel Raïssa Guemdjo Kamdem & Carlos Ougouyandjou, 2020. "S-ARMA model and Wold decomposition for covariance stationary interval-valued time series processes," Post-Print hal-02901595, HAL.
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    Keywords

    Uncertainty Modeling; Stochastic Processes; Random Extended Interval; Time series; ARMA; Hausdorff Metrics;
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