Tail dependence of recursive max-linear models with regularly varying noise variables

Gissibl, Nadine; Klüppelberg, Claudia; Otto, Moritz

doi:10.1016/j.ecosta.2018.02.003

Cited by 16 publications

(9 citation statements)

References 33 publications

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“…Alternative approaches assume regular variation of Z ; see Gissibl et al . () and Klüppelberg and Krali ().…”

Section: Discussion On the Paper By Engelke And Hitzmentioning

confidence: 99%

“…Gissibl and Klüppelberg () and Gissibl et al . () studied the causal structure of directed acyclic graphs for max‐linear models, and they developed methods for model identification based on tail dependence coefficients. Their work is in some sense complementary to ours, since their models do not have densities whereas we shall explicitly assume the existence of densities.…”

Section: Introductionmentioning

confidence: 99%

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Graphical Models for Extremes

Engelke

Hitz

2020

Journal of the Royal Statistical Society Series B: Statistical Methodology

101

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Summary Conditional independence, graphical models and sparsity are key notions for parsimonious statistical models and for understanding the structural relationships in the data. The theory of multivariate and spatial extremes describes the risk of rare events through asymptotically justified limit models such as max‐stable and multivariate Pareto distributions. Statistical modelling in this field has been limited to moderate dimensions so far, partly owing to complicated likelihoods and a lack of understanding of the underlying probabilistic structures. We introduce a general theory of conditional independence for multivariate Pareto distributions that enables the definition of graphical models and sparsity for extremes. A Hammersley–Clifford theorem links this new notion to the factorization of densities of extreme value models on graphs. For the popular class of Hüsler–Reiss distributions we show that, similarly to the Gaussian case, the sparsity pattern of a general extremal graphical model can be read off from suitable inverse covariance matrices. New parametric models can be built in a modular way and statistical inference can be simplified to lower dimensional marginals. We discuss learning of minimum spanning trees and model selection for extremal graph structures, and we illustrate their use with an application to flood risk assessment on the Danube river.

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“…Alternative approaches assume regular variation of Z ; see Gissibl et al . () and Klüppelberg and Krali ().…”

Section: Discussion On the Paper By Engelke And Hitzmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Graphical Models for Extremes

Engelke

Hitz

2020

Journal of the Royal Statistical Society Series B: Statistical Methodology

101

View full text Add to dashboard Cite

show abstract

“…Extreme value models often rely on regular variation and several publications have combined Bayesian networks with such heavy-tailed innovations. In [18] and [21], algorithms have been proposed for statistically learning the model based on the estimated tail dependence matrix and on a scaling method, respectively. In [11] for undirected graphs the authors apply a peaks-over-threshold approach giving a multivariate generalized Pareto distribution for exceedances such that a density exists.…”

Section: Discussionmentioning

confidence: 99%

Conditional Independence in Max-linear Bayesian Networks

Améndola,

Klüppelberg,

Lauritzen

et al. 2020

Preprint

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Motivated by extreme value theory, max-linear Bayesian networks have been recently introduced and studied as an alternative to linear structural equation models. However, for max-linear systems the classical independence results for Bayesian networks are far from exhausting valid conditional independence statements. We use tropical linear algebra to derive a compact representation of the conditional distribution given a partial observation, and exploit this to obtain a complete description of all conditional independence relations. In the context-dependent case, where conditional independence is queried relative to a specific value of the conditioning variables, we introduce the notion of a source DAG to disclose the valid conditional independence relations. In the context-independent case we characterize conditional independence through a modified separation concept, * -separation, combined with a tropical eigenvalue condition. We also introduce the notion of an impact graph which describes how extreme events spread deterministically through the network and we give a complete characterization of such impact graphs. Our analysis opens up several interesting questions concerning conditional independence and tropical geometry.

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“…In [1], a metric that takes the distance along a river into account underlies a spatial model for extremes of river networks. Recursive max-linear models on directed acyclic graphs are proposed in [13] and put to work in [9] and [14]. In [17], the density of a multivariate Pareto distribution is factorized through a version of the Hammersley-Clifford theorem.…”

Section: Introductionmentioning

confidence: 99%

One- versus multi-component regular variation and extremes of Markov trees

Segers

2020

Adv. Appl. Probab.

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A Markov tree is a random vector indexed by the nodes of a tree whose distribution is determined by the distributions of pairs of neighbouring variables and a list of conditional independence relations. Upon an assumption on the tails of the Markov kernels associated to these pairs, the conditional distribution of the self-normalized random vector when the variable at the root of the tree tends to infinity converges weakly to a random vector of coupled random walks called a tail tree. If, in addition, the conditioning variable has a regularly varying tail, the Markov tree satisfies a form of one-component regular variation. Changing the location of the root, that is, changing the conditioning variable, yields a different tail tree. When the tails of the marginal distributions of the conditioning variables are balanced, these tail trees are connected by a formula that generalizes the time change formula for regularly varying stationary time series. The formula is most easily understood when the various one-component regular variation statements are tied up into a single multi-component statement. The theory of multi-component regular variation is worked out for general random vectors, not necessarily Markov trees, with an eye towards other models, graphical or otherwise.

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Tail dependence of recursive max-linear models with regularly varying noise variables

Cited by 16 publications

References 33 publications

Graphical Models for Extremes

Graphical Models for Extremes

Conditional Independence in Max-linear Bayesian Networks

One- versus multi-component regular variation and extremes of Markov trees

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