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Inference on extremal dependence in the domain of attraction of a structured Hüsler–Reiss distribution motivated by a Markov tree with latent variables

Author

Listed:
  • Asenova, Stefka Kirilova

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

  • Mazo, Gildas
  • Segers, Johan

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

Abstract

A Markov tree is a probabilistic graphical model for a random vector indexed by the nodes of an undirected tree encoding conditional independence relations between variables. One possible limit distribution of partial maxima of samples from such a Markov tree is a max-stable Hüsler–Reiss distribution whose parameter matrix inherits its structure from the tree, each edge contributing one free dependence parameter. Our central assumption is that, upon marginal standardization, the data-generating distribution is in the max-domain of attraction of the said Hüsler–Reiss distribution, an assumption much weaker than the one that data are generated according to a graphical model. Even if some of the variables are unobservable (latent), we show that the underlying model parameters are still identifiable if and only if every node corresponding to a latent variable has degree at least three. Three estimation procedures, based on the method of moments, maximum composite likelihood, and pairwise extremal coefficients, are proposed for usage on multivariate peaks over thresholds data when some variables are latent. A typical application is a river network in the form of a tree where, on some locations, no data are available. We illustrate the model and the identifiability criterion on a data set of high water levels on the Seine, France, with two latent variables. The structured Hüsler–Reiss distribution is found to fit the observed extremal dependence patterns well. The parameters being identifiable we are able to quantify tail dependence between locations for which there are no data.

Suggested Citation

  • 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).
  • Handle: RePEc:aiz:louvar:2021004
    DOI: https://doi.org/10.1007/s10687-021-00407-5
    Note: In: Extremes, to appear (2021)
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    Citations

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    Cited by:

    1. Hu, Shuang & Peng, Zuoxiang & Segers, Johan, 2022. "Modelling multivariate extreme value distributions via Markov trees," LIDAM Discussion Papers ISBA 2022021, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Natalia Markovich & Marijus Vaičiulis, 2023. "Extreme Value Statistics for Evolving Random Networks," Mathematics, MDPI, vol. 11(9), pages 1-35, May.
    3. Asenova, Stefka & Segers, Johan, 2022. "Max-linear graphical models with heavy-tailed factors on trees of transitive tournaments," LIDAM Discussion Papers ISBA 2022031, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    4. Sebastian Engelke & Stanislav Volgushev, 2022. "Structure learning for extremal tree models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(5), pages 2055-2087, November.
    5. 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).
    6. Hentschel, Manuel & Engelke, Sebastian & Segers, Johan, 2022. "Statistical Inference for Hüsler–Reiss Graphical Models Through Matrix Completions," LIDAM Discussion Papers ISBA 2022032, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).

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