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Max-linear graphical models with heavy-tailed factors on trees of transitive tournaments

Part of: Probability theory on algebraic and topological structures Stochastic processes

Published online by Cambridge University Press:  15 December 2023

Stefka Asenova  and
Johan Segers Open the ORCID record for Johan Segers [Opens in a new window]
Stefka Asenova*
Affiliation:
UCLouvain
Johan Segers*
Affiliation:
UCLouvain
*
*Postal address: LIDAM/ISBA, Voie du Roman Pays 20, 1348 Louvain-la-Neuve, Belgium.
*Postal address: LIDAM/ISBA, Voie du Roman Pays 20, 1348 Louvain-la-Neuve, Belgium.
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Abstract

Graphical models with heavy-tailed factors can be used to model extremal dependence or causality between extreme events. In a Bayesian network, variables are recursively defined in terms of their parents according to a directed acyclic graph (DAG). We focus on max-linear graphical models with respect to a special type of graph, which we call a tree of transitive tournaments. The latter is a block graph combining in a tree-like structure a finite number of transitive tournaments, each of which is a DAG in which every two nodes are connected. We study the limit of the joint tails of the max-linear model conditionally on the event that a given variable exceeds a high threshold. Under a suitable condition, the limiting distribution involves the factorization into independent increments along the shortest trail between two variables, thereby imitating the behaviour of a Markov random field.

We are also interested in the identifiability of the model parameters in the case when some variables are latent and only a subvector is observed. It turns out that the parameters are identifiable under a criterion on the nodes carrying the latent variables which is easy and quick to check.

Keywords

Max-linear modelheavy tailsextremal dependenceconditional dependenceprobabilistic graphical modeldirected acyclic graphtournamentsextremes

MSC classification

Primary: 60B12: Limit theorems for vector-valued random variables (infinite-dimensional case)
Secondary: 60G70: Extreme value theory; extremal processes
Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 45
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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