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Models for Dependent Extremes Using Stable Mixtures

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  • ANNE‐LAURE FOUGÈRES
  • JOHN P. NOLAN
  • HOLGER ROOTZÉN

Abstract

. This paper unifies and extends results on a class of multivariate extreme value (EV) models studied by Hougaard, Crowder and Tawn. In these models, both unconditional and conditional distributions are themselves EV distributions, and all lower‐dimensional marginals and maxima belong to the class. One interpretation of the models is as size mixtures of EV distributions, where the mixing is by positive stable distributions. A second interpretation is as exponential‐stable location mixtures (for Gumbel) or as power‐stable scale mixtures (for non‐Gumbel EV distributions). A third interpretation is through a peaks over thresholds model with a positive stable intensity. The mixing variables are used as a modelling tool and for better understanding and model checking. We study EV analogues of components of variance models, and new time series, spatial and continuous parameter models for extreme values. The results are applied to data from a pitting corrosion investigation.

Suggested Citation

  • Anne‐Laure Fougères & John P. Nolan & Holger Rootzén, 2009. "Models for Dependent Extremes Using Stable Mixtures," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(1), pages 42-59, March.
    • Handle: RePEc:bla:scjsta:v:36:y:2009:i:1:p:42-59
    DOI: 10.1111/j.1467-9469.2008.00613.x
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    References listed on IDEAS

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    1. Martin Crowder, 1998. "A Multivariate Model for Repeated Failure Time Measurements," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 25(1), pages 53-67, March.
    2. 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.
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    Citations

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    Cited by:

    1. Anthony Medford, 2021. "Modeling Best Practice Life Expectancy Using Gumbel Autoregressive Models," Risks, MDPI, vol. 9(3), pages 1-10, March.
    2. Goix, Nicolas & Sabourin, Anne & Clémençon, Stephan, 2017. "Sparse representation of multivariate extremes with applications to anomaly detection," Journal of Multivariate Analysis, Elsevier, vol. 161(C), pages 12-31.
    3. Robert E. Melchers & Mukshed Ahammed, 2021. "Estimating the Long-Term Reliability of Steel and Cast Iron Pipelines Subject to Pitting Corrosion," Sustainability, MDPI, vol. 13(23), pages 1-10, November.
    4. Leonardo Brandao Freitas Nascimento & Max Sousa Lima & Luiz H. Duczmal, 2025. "P‐min‐Stable Regression Models for Time Series With Extreme Values of Limited Range," Environmetrics, John Wiley & Sons, Ltd., vol. 36(2), March.
    5. Fougères, Anne-Laure & Mercadier, Cécile & Nolan, John P., 2013. "Dense classes of multivariate extreme value distributions," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 109-129.
    6. Kiriliouk, Anna, 2020. "Hypothesis testing for tail dependence parameters on the boundary of the parameter space," Econometrics and Statistics, Elsevier, vol. 16(C), pages 121-135.
    7. Sabourin, Anne, 2015. "Semi-parametric modeling of excesses above high multivariate thresholds with censored data," Journal of Multivariate Analysis, Elsevier, vol. 136(C), pages 126-146.
    8. Samuel A. Morris & Brian J. Reich & Emeric Thibaud, 2019. "Exploration and Inference in Spatial Extremes Using Empirical Basis Functions," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 24(4), pages 555-572, December.
    9. Segers, Johan, 2012. "Max-Stable Models For Multivariate Extremes," LIDAM Discussion Papers ISBA 2012011, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    10. Enkelejd Hashorva & Simone A. Padoan & Stefano Rizzelli, 2021. "Multivariate extremes over a random number of observations," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(3), pages 845-880, September.
    11. Bernhart German & Mai Jan-Frederik & Scherer Matthias, 2015. "On the construction of low-parametric families of min-stable multivariate exponential distributions in large dimensions," Dependence Modeling, De Gruyter, vol. 3(1), pages 1-18, May.
    12. Mai, Jan-Frederik, 2018. "Extreme-value copulas associated with the expected scaled maximum of independent random variables," Journal of Multivariate Analysis, Elsevier, vol. 166(C), pages 50-61.

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