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Extremes of Markov random fields on block graphs

Author

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  • Asenova, Stefka

    (Université catholique de Louvain, LIDAM/ISBA, Belgium)

  • Segers, Johan

    (Université catholique de Louvain, LIDAM/ISBA, Belgium)

Abstract

We study the joint occurrence of large values of a Markov random field or undirected graphical model associated to a block graph. On such graphs, containing trees as specialcases, we aim to generalize recent results for extremes of Markov trees. Every pair ofnodes in a block graph is connected by a unique shortest path. These paths are shownto determine the limiting distribution of the properly rescaled random field given that a fixed variable exceeds a high threshold. When the sub-vectors induced by the blocks follow Hüsler–Reiss extreme value copulas, the global Markov property of the original field induces a particular structure on the parameter matrix of the limiting max-stable Hüsler–Reiss distribution. The multivariate Pareto version of the latter turns out to be an extremal graphical model according to the original block graph. Moreover, thanks to these algebraic relations, the parameters are still identifiable even if some variables are latent.

Suggested Citation

  • Asenova, Stefka & Segers, Johan, 2022. "Extremes of Markov random fields on block graphs," LIDAM Discussion Papers ISBA 2022013, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    • Handle: RePEc:aiz:louvad:2022013
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    References listed on IDEAS

    as
    1. Sebastian Engelke & Adrien S. Hitz, 2020. "Graphical models for extremes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(4), pages 871-932, September.
    2. Rootzen, Holger & Segers, Johan & Wadsworth, Jennifer L., 2018. "Multivariate generalized Pareto distributions: Parametrizations, representations, and properties," LIDAM Reprints ISBA 2018003, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    3. Asenova, Stefka Kirilova & Mazo, Gildas & Segers, Johan, 2021. "Inference on extremal dependence in the domain of attraction of a structured Hüsler–Reiss distribution motivated by a Markov tree with latent variables," LIDAM Reprints ISBA 2021004, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
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    7. Sebastian Engelke & Alexander Malinowski & Zakhar Kabluchko & Martin Schlather, 2015. "Estimation of Hüsler–Reiss distributions and Brown–Resnick processes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 77(1), pages 239-265, January.
    8. Rootzén, Holger & Segers, Johan & Wadsworth, Jennifer L., 2018. "Multivariate generalized Pareto distributions: Parametrizations, representations, and properties," Journal of Multivariate Analysis, Elsevier, vol. 165(C), pages 117-131.
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