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An abelian way approach to study random extended intervals and their ARMA processes

Author

Listed:
  • Babel Raïssa Guemdjo Kamdem

    (Université de Douala)

  • Jules Sadefo Kamdem

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

  • Carlos Ogouyandjou

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

Abstract

An extended interval is a range A = [A, A] where A may be bigger than A. This is not really natural, but is what has been used as the definition of an extended interval so far. In the present work we introduce a new, natural, and very intuitive way to see an extended interval. From now on, an extended interval is a subset of the Cartesian product R × Z2, where Z2 = {0, 1} is the set of directions; the direction 0 is for increasing intervals, and the direction 1 for decreasing ones. For instance, [3, 6] × {1} is the decreasing version of [6, 3]. Thereafter, we introduce on the set of extended intervals a family of metrics dγ, depending on a function γ(t), and show that there exists a unique metric dγ for which γ(t)dt is what we have called an "adapted measure". This unique metric has very good properties, is simple to compute, and has been implemented in the software R. Furthermore, we use this metric to define variability for random extended intervals. We further study extended interval-valued ARMA time series and prove the Wold decomposition theorem for stationary extended interval-valued times series.

Suggested Citation

  • Babel Raïssa Guemdjo Kamdem & Jules Sadefo Kamdem & Carlos Ogouyandjou, 2024. "An abelian way approach to study random extended intervals and their ARMA processes," Post-Print hal-04506343, HAL.
  • Handle: RePEc:hal:journl:hal-04506343
    DOI: 10.3934/dsfe.2024005
    Note: View the original document on HAL open archive server: https://hal.science/hal-04506343v1
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    References listed on IDEAS

    as
    1. 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.
    2. 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.
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    More about this item

    Keywords

    Random set; Random extended interval; Distance; Measure; Time series;
    All these keywords.

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