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

Abstract

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.

A flexible dependence model for spatial extremes

“…Recently, improvements have been obtained for the parameters estimations of some maxstable processes, e.g. 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.…”

Geostatistics of Dependent and Asymptotically Independent Extremes

“…Brown-Resnick processes can be shown to be essentially the only limit of properly rescaled maxima of Gaussian processes (Kabluchko et al 2009) and have been found to yield good fits in applications Jeon and Smith 2012). Their flexibility, and the recent development of efficient inference procedures (Engelke et al 2012;Wadsworth and Tawn 2013), make them particularly attractive. Although Brown-Resnick processes are based on nonstationary Gaussian models, the construction in Eq.…”

“…Since the max-stable models are suitable only above some predetermined high threshold, inference is usually made using a censored approach (Huser andJeon and Smith 2012;Thibaud et al 2013). Furthermore, following Stephenson and Tawn (2005), Davison and Gholamrezaee (2012) and Wadsworth and Tawn (2013) show how to incorporate the occurrence times of extreme events, use of which both simplifies the likelihood and allows much more efficient inference in cases of moderate to low spatial dependence.…”

Trend in the Co‐Occurrence of Extreme Daily Rainfall in West Africa Since 1950

“…Assuming temporal independence of the y 's, but spatial dependence, the log‐likelihood of the model is written as where g is the multivariate density of Brown‐Resnick model. Wadsworth and Tawn () give a closed form expression for g ; however, its computation results in a combinatorial explosion (Castruccio et al, ; Davison & Gholamrezaee, ). It is possible to circumvent this issue by making estimation based on the pairwise log‐likelihood (Padoan et al, ; Varin et al, ) where g i j is the bivariate density of ( Y i , Y j ), that is, associated to , and N is the number of stations.…”