The generalized Pareto process; with a view towards application and simulation

Ferreira, Ana; Haan, Leo de

doi:10.3150/13-bej538

Cited by 108 publications

(130 citation statements)

References 11 publications

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“…Brown-Resnick processes: extremal increments of the process allow to work with a complete likelihood function (Engelke et al, 2015;Wadsworth and Tawn, 2014). A direct modeling of the exceedances of a max-stable process is also possible using a generalized Pareto process (Ferreira and de Haan, 2014) but such an approach is only of interest in the case of asymptotic dependence.…”

Section: Model Inferencementioning

confidence: 99%

A flexible dependence model for spatial extremes

Bacro

Gaetan

Toulemonde

2016

Journal of Statistical Planning and Inference

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Max-stable processes play a fundamental role in modeling the spatial dependence of extremes because they appear as a natural extension of multivariate extreme value distributions. In practice, a well-known restrictive assumption when using max-stable processes comes from the fact that the observed extremal dependence is assumed to be related to a particular max-stable dependence structure. As a consequence, the latter is imposed to all events which are more extreme than those that have been observed. Such an assumption is inappropriate in the case of asymptotic independence. Following recent advances in the literature, we exploit a max-mixture model to suggest a general spatial model which ensures extremal dependence at small distances, possible independence at large distances and asymptotic independence at intermediate distances.Parametric inference is carried out using a pairwise composite likelihood approach. Finally we apply our modeling framework to analyze daily precipitations over the East of Australia, using block maxima over the observation period and exceedances over a large threshold.2

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Section: Model Inferencementioning

confidence: 99%

A flexible dependence model for spatial extremes

Bacro

Gaetan

Toulemonde

2016

Journal of Statistical Planning and Inference

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“…de Haan and Lin (2001), Part III of de Haan and Ferreira (2006), Einmahl and Lin (2006) and Ferreira and de Haan (2014). For the theory presented here, the main difference between the R m setting and the C b (K) setting is that in the latter, exponential tightness of {P (Y/y ∈ ·), y > 0} no longer follows from the exponential marginals; it is an independent assumption.…”

Section: Discussionmentioning

confidence: 99%

Approximation and estimation of very small probabilities of multivariate extreme events

Valk

2016

Extremes

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This article discusses modelling of the tail of a multivariate distribution function by means of a large deviation principle (LDP), and its application to the estimation of the probability p n of a multivariate extreme event from a sample of n iid random vectors, with p n ∈ [n −τ 2 , n −τ 1 ] for some τ 1 > 1 and τ 2 > τ 1 . One way to view the classical tail limits is as limits of probability ratios. In contrast, the tail LDP provides asymptotic bounds or limits for log-probability ratios. After standardising the marginals to standard exponential, tail dependence is represented by a homogeneous rate function I . Furthermore, the tail LDP can be extended to represent both dependence and marginals, the latter implying marginal log-Generalised Weibull tail limits. A connection is established between the tail LDP and residual tail dependence (or hidden regular variation) and a recent extension of it. Under a smoothness assumption, they are implied by the tail LDP. Based on the tail LDP, a simple estimator for very small probabilities of extreme events is formulated. It avoids estimation of I by making use of its homogeneity. Strong consistency in the sense of convergence of log-probability ratios is proven. Simulations and an application illustrate the difference between the classical approach and the LDP-based approach.

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“…A generalized Pareto process is defined as μ(s ) + σ (s )[Y ∞ (s ) ξ (s ) − 1]/ξ (s ) for the spatially varying location, scale, and shape parameters μ(s ), σ (s ) > 0, and ξ (s ), respectively; the case ξ (s ) = 0 is treated as ξ (s ) → 0. Ferreira & de Haan (2014) have shown that these processes arise as limits as n → ∞ for the rescaled process [Y (s ) − b n (s )]/a n (s ), conditional on there being at least one exceedance of some threshold u. Therefore, unlike max-stable processes, they provide a valid approximation to the upper tail when at least one variable is large.…”

Section: Pareto Processesmentioning

confidence: 95%

“…This approximation is justified when all variables are large; when not all variables are large, a censored likelihood approach may be adopted (see, for example, Huser & Davison 2014). The second is to use Pareto processes (Ferreira & de Haan 2014, Thibaud & Opitz 2014, recently introduced as the process analog of the GPD. In the same vein as Equation 25, a standard Pareto process can be defined as follows:…”

Section: Pareto Processesmentioning

confidence: 99%

Statistics of Extremes

Davison

Huser

2015

Annu. Rev. Stat. Appl.

166

126

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Statistics of extremes concerns inference for rare events. Often the events have never yet been observed, and their probabilities must therefore be estimated by extrapolation of tail models fitted to available data. Because data concerning the event of interest may be very limited, efficient methods of inference play an important role. This article reviews this domain, emphasizing current research topics. We first sketch the classical theory of extremes for maxima and threshold exceedances of stationary series. We then review multivariate theory, distinguishing asymptotic independence and dependence models, followed by a description of models for spatial and spatiotemporal extreme events. Finally, we discuss inference and describe two applications. Animations illustrate some of the main ideas. 9.1

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The generalized Pareto process; with a view towards application and simulation

Cited by 108 publications

References 11 publications

A flexible dependence model for spatial extremes

A flexible dependence model for spatial extremes

Approximation and estimation of very small probabilities of multivariate extreme events

Statistics of Extremes

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