Extremes of Markov random fields on block graphs: max-stable limits and structured Hüsler–Reiss distributions

Stefka Asenova and Johan Segers ()
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Stefka Asenova: Université catholique de Louvain, LIDAM/ISBA, Belgium
Johan Segers: Université catholique de Louvain, LIDAM/ISBA, Belgium

No 2023013, LIDAM Reprints ISBA from Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA)

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 special cases, we aim to generalize recent results for extremes of Markov trees. Every pair of nodes in a block graph is connected by a unique shortest path. These paths are shown to determine the limiting distribution of the properly rescaled random field given that a fixed variable exceeds a high threshold. The latter limit relation implies that the random field is multivariate regularly varying and it determines the max-stable distribution to which component-wise maxima of independent random samples from the field are attracted. When the sub-vectors induced by the blocks have certain limits parametrized by Hüsler–Reiss distributions, 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. Thanks to these algebraic relations, the parameters are still identifiable even if some variables are latent.

Keywords: Markov random field; graphical model; block graph; multivariate extremes; tail dependence; latent variable; Hüsler–Reiss distribution; conditional independence (search for similar items in EconPapers)
Pages: 29
Date: 2023-04-04
Note: In: Extremes, 2023
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Persistent link: https://EconPapers.repec.org/RePEc:aiz:louvar:2023013

DOI: 10.1007/s10687-023-00467-9

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