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Temporal evolution of the extreme excursions of multivariate k$$ k $$th order Markov processes with application to oceanographic data

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  • Stan Tendijck
  • Philip Jonathan
  • David Randell
  • Jonathan Tawn

Abstract

We develop two models for the temporal evolution of extreme events of multivariate k$$ k $$th order Markov processes. The foundation of our methodology lies in the conditional extremes model of Heffernan and Tawn (Journal of the Royal Statistical Society: Series B (Methodology), 2014, 66, 497–546), and it naturally extends the work of Winter and Tawn (Journal of the Royal Statistical Society: Series C (Applied Statistics), 2016, 65, 345–365; Extremes, 2017, 20, 393–415) and Tendijck et al. (Environmetrics 2019, 30, e2541) to include multivariate random variables. We use cross‐validation‐type techniques to develop a model order selection procedure, and we test our models on two‐dimensional meteorological‐oceanographic data with directional covariates for a location in the northern North Sea. We conclude that the newly‐developed models perform better than the widely used historical matching methodology for these data.

Suggested Citation

  • Stan Tendijck & Philip Jonathan & David Randell & Jonathan Tawn, 2024. "Temporal evolution of the extreme excursions of multivariate k$$ k $$th order Markov processes with application to oceanographic data," Environmetrics, John Wiley & Sons, Ltd., vol. 35(3), May.
    • Handle: RePEc:wly:envmet:v:35:y:2024:i:3:n:e2834
    DOI: 10.1002/env.2834
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    References listed on IDEAS

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    1. Paola Bortot & Stuart Coles, 2003. "Extremes of Markov chains with tail switching potential," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(4), pages 851-867, November.
    2. Keef, Caroline & Papastathopoulos, Ioannis & Tawn, Jonathan A., 2013. "Estimation of the conditional distribution of a multivariate variable given that one of its components is large: Additional constraints for the Heffernan and Tawn model," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 396-404.
    3. Stuart G. Coles & Jonathan A. Tawn, 1994. "Statistical Methods for Multivariate Extremes: An Application to Structural Design," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 43(1), pages 1-31, March.
    4. S. Tendijck & E. Ross & D. Randell & P. Jonathan, 2019. "A model for the directional evolution of severe ocean storms," Environmetrics, John Wiley & Sons, Ltd., vol. 30(1), February.
    5. V. Chavez‐Demoulin & A. C. Davison, 2005. "Generalized additive modelling of sample extremes," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 54(1), pages 207-222, January.
    6. David Randell & Graham Feld & Kevin Ewans & Philip Jonathan, 2015. "Distributions of return values for ocean wave characteristics in the South China Sea using directional–seasonal extreme value analysis," Environmetrics, John Wiley & Sons, Ltd., vol. 26(6), pages 442-450, September.
    7. Luc Bauwens & Sébastien Laurent & Jeroen V. K. Rombouts, 2006. "Multivariate GARCH models: a survey," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 21(1), pages 79-109, January.
    8. Axel Gandy & Kaushik Jana & Almut E. D. Veraart, 2022. "Scoring predictions at extreme quantiles," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 106(4), pages 527-544, December.
    9. Anthony W. Ledford & Jonathan A. Tawn, 2003. "Diagnostics for dependence within time series extremes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(2), pages 521-543, May.
    10. Hilal, Sawsan & Poon, Ser-Huang & Tawn, Jonathan, 2011. "Hedging the black swan: Conditional heteroskedasticity and tail dependence in S&P500 and VIX," Journal of Banking & Finance, Elsevier, vol. 35(9), pages 2374-2387, September.
    11. Janet E. Heffernan & Jonathan A. Tawn, 2004. "A conditional approach for multivariate extreme values (with discussion)," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(3), pages 497-546, August.
    12. Emma F. Eastoe & Jonathan A. Tawn, 2012. "Modelling the distribution of the cluster maxima of exceedances of subasymptotic thresholds," Biometrika, Biometrika Trust, vol. 99(1), pages 43-55.
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