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

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 distributio...

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Bibliographic Details
Published in:Extremes (Boston) 2021-09, Vol.24 (3), p.461-500
Main Authors: Asenova, Stefka, Mazo, Gildas, Segers, Johan
Format: Article
Language:English
Subjects:
Attraction
Civil Engineering
Domains
Economics
Environmental Management
Finance
Hydrogeology
Insurance
Management
Mathematical models
Mathematics
Mathematics and Statistics
Maxima
Method of moments
Multivariate analysis
Parameter identification
Quality Control
Reliability
River networks
Safety and Risk
Standardization
Statistics
Statistics for Business
Variables
Water levels
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Summary: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.
ISSN:1386-1999
1572-915X
DOI:10.1007/s10687-021-00407-5