On the structure and representations of max-stable processes

Wang, Yizao; Stoev, Stilian

doi:10.1017/s0001867800050473

Cited by 46 publications

(79 citation statements)

References 33 publications

(59 reference statements)

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“…The extension to the multiparameter case where T ≠ m is a simple generalization using theorem 6.1.2 in Krengel (), which is a multiparameter extension of the Ackoglus ergodic theorem. Ergodic properties of Brown–Resnick processes have been studied for the uniparameter case in Stoev and Taqqu () and Wang and Stoev (). The Brown–Resnick process (2) has a stochastic representation

2.047em2.047emtrue{\int_{E}^{e} exp false{W (s, t) - italicδ (s, t) false} d M_{1}, s \in R^{d}, t \in false[0, \infty false) 2.047em2.047emtrue},

where

M_{1}

is a random 1‐Fréchet sup‐measure on the probability space

(Ω, E, P)

on which the Gaussian process W is defined.…”

Section: Strong Consistency Of the Pairwise Likelihood Estimates For mentioning

confidence: 99%

Statistical Inference for Max-Stable Processes in Space and Time

Davis

Klüppelberg

Steinkohl

2013

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Summary Max‐stable processes have proved to be useful for the statistical modelling of spatial extremes. Several families of max‐stable random fields have been proposed in the literature. One such representation is based on a limit of normalized and rescaled pointwise maxima of stationary Gaussian processes that was first introduced by Kabluchko and co‐workers. This paper deals with statistical inference for max‐stable space–time processes that are defined in an analogous fashion. We describe pairwise likelihood estimation, where the pairwise density of the process is used to estimate the model parameters. For regular grid observations we prove strong consistency and asymptotic normality of the parameter estimates as the joint number of spatial locations and time points tends to ∞. Furthermore, we discuss extensions to irregularly spaced locations. A simulation study shows that the method proposed works well for these models.

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2.047em2.047emtrue{\int_{E}^{e} exp false{W (s, t) - italicδ (s, t) false} d M_{1}, s \in R^{d}, t \in false[0, \infty false) 2.047em2.047emtrue},

where

M_{1}

is a random 1‐Fréchet sup‐measure on the probability space

(Ω, E, P)

on which the Gaussian process W is defined.…”

Section: Strong Consistency Of the Pairwise Likelihood Estimates For mentioning

confidence: 99%

Statistical Inference for Max-Stable Processes in Space and Time

Davis

Klüppelberg

Steinkohl

2013

Journal of the Royal Statistical Society Series B: Statistical Methodology

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“…The model simplifies for β = α to the power model of a fractional Brownian random field (Mandelbrot and Van Ness, ), for 0< β ⩽ α to a generalization of the fractional Brownian model described in the work of Schlather (), and for β <0 to the generalized Cauchy model (Gneiting, ; Gneiting and Schlather, ). It also generalizes some of the multiquadric and inverse multiquadric models used in approximation theory where α =2 and

β / α \in double-struckR ∖ {double-struckN}_{0}

, see, for instance, Buhmann (), Wendland () or Lin and Yuan (). As

β \to 0

, the limiting model equals a modified version of the De Wijsian model (Wackernagel, ; Matheron, ).…”

Section: The Modelmentioning

confidence: 85%

A parametric model bridging between bounded and unbounded variograms

Schlather

Moreva

2017

Stat

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A simple variogram model with two parameters is presented that includes the power variogram for fractional Brownian motion, a modified De Wijsian model, the generalized Cauchy model and the multiquadric model. One parameter controls the sample path roughness of the process. The other parameter allows for a smooth transition between bounded and unbounded variograms, that is, between stationary and intrinsically stationary processes in a Gaussian framework, or between mixing and non-ergodicBrown-Resnick processes when modeling spatial extremes. Stat 2017; 6: 47-52 Stat Bridging between bounded and unbounded variograms The ISI's Journal for the Rapid (wileyonlinelibrary.com)

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“…For Gaussian process models, spatial prediction is accomplished through the classical geostatistical technique Kriging. Under previous max‐stable process models, spatial prediction is possible but complicated (e.g., Wang and Stoev, ). The HKEVP, in contrast, permits straightforward spatial prediction at unobserved sites.…”

Section: A Spatial Max‐stable Process Modelmentioning

confidence: 99%

Bayesian spatial extreme value analysis to assess the changing risk of concurrent high temperatures across large portions of European cropland

Shaby

Reich

2012

Environmetrics

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There is strong evidence that extremely high temperatures are detrimental to the yield and quality of many economically and socially critical crops. Fortunately, the most deleterious conditions for agriculture occur rarely. We wish to assess the risk of the catastrophic scenario in which large areas of croplands experience extreme heat stress during the same growing season. Applying a hierarchical Bayesian spatial extreme value model that allows the distribution of extreme temperatures to change in time both marginally and in spatial coherence, we examine whether the risk of widespread extremely high temperatures across agricultural land in Europe has increased over the last century. Copyright © 2012 John Wiley & Sons, Ltd.

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On the structure and representations of max-stable processes

Cited by 46 publications

References 33 publications

Statistical Inference for Max-Stable Processes in Space and Time

Statistical Inference for Max-Stable Processes in Space and Time

A parametric model bridging between bounded and unbounded variograms

Bayesian spatial extreme value analysis to assess the changing risk of concurrent high temperatures across large portions of European cropland

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