Extreme values of independent stochastic processes

Brown, B. M.; Resnick, Sidney I.

doi:10.1017/s0021900200105261

Cited by 129 publications

(207 citation statements)

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“…Thus pairwise likelihood inference provides a good compromise between statistical and computational efficiency in many applications. (Brown and Resnick, 1977;Kabluchko et al, 2009) is a stationary max-stable process that may be represented as Z(x) = sup i∈N W i (x)/T i (x ∈ X ⊂ R d ), where 0 < T 1 < T 2 < · · · are the points of a unit rate Poisson process on R + and the W i (x) are independent replicates of the random process W (x) = exp{ε(x) − γ(x)}. Here ε(x) is an intrinsically stationary Gaussian random field with semi-variogram γ(h) with ε(0) = 0 almost surely.…”

Section: Introductionmentioning

confidence: 99%

Composite likelihood estimation for the Brown-Resnick process

Huser¹,

Davison²

2013

Biometrika

106

105

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SUMMARY Genton et al. (2011) investigated the gain in efficiency when triplewise, rather than pairwise, likelihood is used to fit the popular Smith max-stable model for spatial extremes. We generalize their results to the Brown-Resnick model and show that the efficiency gain is substantial only for very smooth processes, which are generally unrealistic in applications.Some key words: Brown-Resnick process; Composite likelihood; Max-stable process; Pairwise likelihood; Smith process; Triplewise likelihood. INTRODUCTIONMax-stable processes are useful for the statistical modelling of spatial extreme events. No finite parametrization of such processes exists, but a spectral representation (de Haan, 1984) aids in constructing models. In a 1990 University of Surrey technical report, R. L. Smith proposed a max-stable model based on deterministic storm profiles which has become popular because it is simple, readily interpreted and easily simulated, but unfortunately it is too inflexible for realism in practice. Another popular model, the Brown-Resnick process, is based on intrinsically stationary log-Gaussian processes, can handle a wide range of dependence structures, and often provides a better fit to data; see, for example, or a 2012 University of North Carolina at Chapel Hill PhD thesis by Soyoung Jeon. Kabluchko et al. (2009) provided further underpinning for this process by showing that that under mild conditions, the BrownResnick process with variogram 2γ(h) = ( h /ρ) α (ρ > 0, 0 < α ≤ 2), where h is the spatial lag, is essentially the only isotropic limit of properly rescaled maxima of Gaussian processes. The Smith model is obtained by taking a Brown-Resnick process with variogram 2γ(h) = h T Σ −1 h for some covariance matrix Σ, corresponding after an affine transformation to taking α = 2, whereas found that 1/2 < α < 1 for the rainfall data they examined.Likelihood inference for max-stable models is difficult, since only the bivariate marginal density functions are known in most cases, and pairwise marginal likelihood is typically used (Padoan et al., 2010;Davison and Gholamrezaee, 2012;Huser and Davison, 2012). This raises the question whether some other approach to inference would be preferable. Genton et al. (2011) derived the general form of the likelihood function for the Smith model and showed that large efficiency gains can arise when fitting it using triplewise, rather than pairwise, likelihood. In this paper we extend their investigation to the Brown-Resnick process and show that for rougher models, more realistic than those considered by Genton et al. (2011), the efficiency gains are much less striking. Thus pairwise likelihood inference provides a good compromise between statistical and computational efficiency in many applications.

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Section: Introductionmentioning

confidence: 99%

Composite likelihood estimation for the Brown-Resnick process

Huser¹,

Davison²

2013

Biometrika

106

105

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“…From these two figures, it is possible to roughly order the five spatial extreme value models by their performances over the two criteria defined in Section 3.2.2. The two "best" competitors may be the BRP of Brown and Resnick (1977); Kabluchko et al (2009) and the ETP of Opitz (2013). Both methods show similar satisfying results in terms of the two criteria of comparison, with an exception on Figure 6b when the generator is the HKEVP.…”

Section: Resultsmentioning

confidence: 86%

“…Then, representation (3) leads to a process Z(·) originally introduced by Brown and Resnick (1977), see also Kabluchko et al (2009). This latter process is traditionally called the geometric Gaussian process or the Brown-Resnick Process, and will be denoted by BRP in the following.…”

Section: The Brown-resnick Process: Brpmentioning

confidence: 99%

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Modeling extreme rainfall A comparative study of spatial extreme value models

2017

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In this paper, focus is done on spatial models for extreme events and on their respective efficiency regarding the estimation of two risk measures: one extrapolating marginal distributions and one summarizing the spatial bivariate dependence of extremes. A wide comparison is performed on an innovative simulation plan that has been specifically designed from a daily precipitation data set. The objective of this paper is twofold: firstly, pointing out the inherent properties of each model, and secondly, advising users on how to choose the model depending on the specific type of risk.

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“…The focus of this paper is inference for a class of processes, commonly known as Brown-Resnick processes (Brown & Resnick, 1977;Kabluchko et al, 2009), whose spectral functions are log-Gaussian random fields. The d-dimensional distribution functions for all such processes are known (Genton et al, 2011;Huser & Davison, 2013;Engelke et al, 2014), but owing to the exponential form of distribution function (4), even when the expectation is calculable, differentiation to yield high-dimensional densities produces an explosion of terms in the density: the d-variate density consists of B d summands, where B d is the dth Bell number.…”

Section: Introductionmentioning

confidence: 99%

Efficient inference for spatial extreme value processes associated to log-Gaussian random functions

2013

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SUMMARYMax-stable processes arise as the only possible nontrivial limits for maxima of affinely normalized identically distributed stochastic processes, and thus form an important class of models for the extreme values of spatial processes. Until recently, inference for max-stable processes has been restricted to the use of pairwise composite likelihoods, due to intractability of higherdimensional distributions. In this work we consider random fields that are in the domain of attraction of a widely used class of max-stable processes, namely those constructed via manipulation of log-Gaussian random functions. For this class, we exploit limiting d-dimensional multivariate Poisson process intensities of the underlying process for inference on all d-vectors exceeding a high marginal threshold in at least one component, employing a censoring scheme to incorporate information below the marginal threshold. We also consider the d-dimensional distributions for the equivalent max-stable process, and perform full likelihood inference by exploiting the methods of Stephenson & Tawn (2005), where information on the occurrence times of extreme events is shown to dramatically simplify the likelihood. The Stephenson-Tawn likelihood is in fact simply a special case of the censored Poisson process likelihood. We assess the improvements in inference from both methods over pairwise likelihood methodology by simulation.

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Extreme values of independent stochastic processes

Cited by 129 publications

References 6 publications

Composite likelihood estimation for the Brown-Resnick process

Composite likelihood estimation for the Brown-Resnick process

Modeling extreme rainfall A comparative study of spatial extreme value models

Efficient inference for spatial extreme value processes associated to log-Gaussian random functions

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