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Extremes of multivariate ARMAX processes

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  • Marta Ferreira
  • Helena Ferreira

Abstract

We define a new multivariate time series model by generalizing the ARMAX process in a multivariate way. We give conditions on stationarity and analyze local dependence and domains of attraction. As a consequence of the obtained results, we derive new multivariate extreme value distributions. We characterize the extremal dependence by computing the multivariate extremal index and bivariate upper tail dependence coefficients. An estimation procedure for the multivariate extremal index is presented. We also address the marginal estimation and propose a new estimator for the ARMAX autoregressive parameter. Copyright Sociedad de Estadística e Investigación Operativa 2013

Suggested Citation

  • Marta Ferreira & Helena Ferreira, 2013. "Extremes of multivariate ARMAX processes," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(4), pages 606-627, November.
    • Handle: RePEc:spr:testjl:v:22:y:2013:i:4:p:606-627
    DOI: 10.1007/s11749-013-0326-6
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    References listed on IDEAS

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    1. Rafael Schmidt & Ulrich Stadtmüller, 2006. "Non‐parametric Estimation of Tail Dependence," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(2), pages 307-335, June.
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    6. Zhengjun Zhang, 2008. "The estimation of M4 processes with geometric moving patterns," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(1), pages 121-150, March.
    7. Einmahl, J.H.J. & Krajina, A. & Segers, J., 2011. "An M-Estimator for Tail Dependence in Arbitrary Dimensions," Discussion Paper 2011-013, Tilburg University, Center for Economic Research.
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    Cited by:

    1. Manuel G. Scotto & Christian H. Weiß & Tobias A. Möller & Sónia Gouveia, 2018. "The max-INAR(1) model for count processes," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(4), pages 850-870, December.
    2. Withers, Christopher S. & Nadarajah, Saralees, 2014. "The distribution of the maximum of the multivariate AR(p) and multivariate MA(p) processes," Statistics & Probability Letters, Elsevier, vol. 95(C), pages 48-56.
    3. Gloria Buriticá & Philippe Naveau, 2023. "Stable sums to infer high return levels of multivariate rainfall time series," Environmetrics, John Wiley & Sons, Ltd., vol. 34(4), June.
    4. Ferreira, Helena & Ferreira, Marta, 2015. "Extremes of scale mixtures of multivariate time series," Journal of Multivariate Analysis, Elsevier, vol. 137(C), pages 82-99.
    5. Helena Ferreira & Ana Paula Martins & Maria Graça Temido, 2021. "Extremal behaviour of a periodically controlled sequence with imputed values," Statistical Papers, Springer, vol. 62(6), pages 2991-3013, December.

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