Multivariate generalized Pareto distributions: Parametrizations, representations, and properties

Rootzén, Holger; Segers, Johan; Wadsworth, Jennifer L.

doi:10.1016/j.jmva.2017.12.003

Cited by 43 publications

(34 citation statements)

References 20 publications

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“…displays

χ_{scriptC} false(q false) = Pr false{F_{i} (X_{i}) > q \forall i \in C false} / false(1 - q false)

for three sets with | C |=2, and C ={1,…,31}. For a multivariate Pareto distribution

χ_{scriptC} false(q false) \equiv χ_{scriptC}

for q sufficiently large (Rootzén et al ., 2018a), and the bivariate estimates from the fitted model in the paper are displayed. For model (53) with k ∈ C and

{truemin}_{j \in C} false{α_{j}^{k} false} < 1

, χ C ( q )↘0 as q →1.…”

Section: Discussion On the Paper By Engelke And Hitzmentioning

confidence: 99%

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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“…displays

χ_{scriptC} false(q false) = Pr false{F_{i} (X_{i}) > q \forall i \in C false} / false(1 - q false)

for three sets with | C |=2, and C ={1,…,31}. For a multivariate Pareto distribution

χ_{scriptC} false(q false) \equiv χ_{scriptC}

for q sufficiently large (Rootzén et al ., 2018a), and the bivariate estimates from the fitted model in the paper are displayed. For model (53) with k ∈ C and

{truemin}_{j \in C} false{α_{j}^{k} false} < 1

, χ C ( q )↘0 as q →1.…”

Section: Discussion On the Paper By Engelke And Hitzmentioning

confidence: 99%

Graphical Models for Extremes

Engelke

Hitz

2020

Journal of the Royal Statistical Society Series B: Statistical Methodology

101

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“…The root change formula in Corollary 3 extends the time change formula for regularly varying stationary time series stemming from [2] and studied extensively in [7] and [20]. Multivariate Pareto distributions as in Theorem 3 are foreshadowed in [29,Section 6.3]; they appear in [11] and [31] when ρ(x) = max (x 1 , . .…”

Section: Regular Variationmentioning

confidence: 94%

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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“…For comparison, generalized Pareto processes have χ u ≡ χ for all u above a certain level (Rootzén et al, 2017), whilst all max-stable processes have χ u − χ (1 − u), as u → 1.…”

Section: Further Dependence Properties Under Asymptotic Dependencementioning

confidence: 99%

“…where the argument s j denotes the jth spatial location; if the data follow a generalized Pareto process law, then this function should be constant as the quantile u tends to one (Rootzén et al, 2017). For environmental data in particular, it is much more common to see estimates of (2) decreasing as u → 1, indicating that dependence weakens with level of extremeness.…”

Section: Introductionmentioning

confidence: 99%

Modeling Spatial Processes with Unknown Extremal Dependence Class

Huser

Wadsworth

2018

Journal of the American Statistical Association

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121

176

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Many environmental processes exhibit weakening spatial dependence as events become more extreme. Well-known limiting models, such as max-stable or generalized Pareto processes, cannot capture this, which can lead to a preference for models that exhibit a property known as asymptotic independence. However, weakening dependence does not automatically imply asymptotic independence, and whether the process is truly asymptotically (in)dependent is usually far from clear. The distinction is key as it can have a large impact upon extrapolation, i.e., the estimated probabilities of events more extreme than those observed. In this work, we present a single spatial model that is able to capture both dependence classes in a parsimonious manner, and with a smooth transition between the two cases. The model covers a wide range of possibilities from asymptotic independence through to complete dependence, and permits weakening dependence of extremes even under asymptotic dependence. Censored likelihood-based inference for the implied copula is feasible in moderate dimensions due to closed-form margins. The model is applied to oceanographic datasets with ambiguous true limiting dependence structure.

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Multivariate generalized Pareto distributions: Parametrizations, representations, and properties

Cited by 43 publications

References 20 publications

Graphical Models for Extremes

Graphical Models for Extremes

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

Modeling Spatial Processes with Unknown Extremal Dependence Class

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