Sebastian Engelke

EXTREMES ON RIVER NETWORKS

Max-stable processes are the natural extension of the classical extreme-value distributions to the functional setting, and they are increasingly widely used to estimate probabilities of complex extreme events. In this paper we broaden them from the usual situation in which dependence varies according to functions of Euclidean distance to situations in which extreme river discharges at two locations on a river network may be dependent because the locations are flow-connected or because of common meteorological events. In the former case dependence depends on river distance, and in the second it depends on the hydrological distance between the locations, either of which may be very different from their Euclidean distance. Inference for the model parameters is performed using a multivariate threshold likelihood, which is shown by simulation to work well. The ideas are illustrated with data from the upper Danube basin.

A unifying approach to fractional Lévy processes

Starting from the moving average representation of fractional Brownian motion fractional L evy processes have been constructed by keeping the same moving average kernel and replacing the Brownian motion by a pure jump L evy process with nite second moments. Another way was to replace the Brownian motion by an alpha-stable L evy process and the exponent in the kernel by H 1= . We now provide a unifying approach taking kernels of the form a((t s) + ( s) +) +b((t s) ( s) ), where can be chosen according to the existing moments and the Blumenthal-Getoor index of the underlying L evy process. These processes may exhibit both long and short range dependence. In addition we will examine further properties of the processes, e.g. regularity of the sample paths and the semimartingale property.

Statistical inference for max-stable processes by conditioning on extreme events

In this paper we provide the basis for new methods of inference for max-stable processes ξ on general spaces that admit a certain incremental representation, which, in important cases, has a much simpler structure than the max-stable process itself. A corresponding peaks-over-threshold approach will incorporate all single events that are extreme in some sense and will therefore rely on a substantially larger amount of data in comparison to estimation procedures based on block maxima. Conditioning a process η in the max-domain of attraction of ξ on being extremal, several convergence results for the increments of η are proved. In a similar way, the shape functions of mixed moving maxima (M3) processes can be extracted from suitably conditioned single events η. Connecting the two approaches, transformation formulae for processes that admit both an incremental and an M3 representation are identified.

Bayesian inference for multivariate extreme value distributions

Abstract: Statistical modeling of multivariate and spatial extreme events has attracted broad attention in various areas of science. Max-stable distributions and processes are the natural class of models for this purpose, and many parametric families have been developed and successfully applied. Due to complicated likelihoods, the efficient statistical inference is still an active area of research, and usually composite likelihood methods based on bivariate densities only are used. Thibaud et al. [2016, Ann. Appl. Stat., to appear] use a Bayesian approach to fit a Brown–Resnick process to extreme temperatures. In this paper, we extend this idea to a methodology that is applicable to general max-stable distributions and that uses full likelihoods. We further provide simple conditions for the asymptotic normality of the median of the posterior distribution and verify them for the commonly used models in multivariate and spatial extreme value statistics. A simulation study shows that this point estimator is considerably more efficient than the composite likelihood estimator in a frequentist framework. From a Bayesian perspective, our approach opens the way for new techniques such as Bayesian model comparison in multivariate and spatial extremes.