Max-stable processes are increasingly widely used for modelling complex extreme events, but existing fitting methods are computationally demanding, limiting applications to a few dozen variables. r-Pareto processes are mathematically simpler and have the potential advantage of incorporating all relevant extreme events, by generalizing the notion of a univariate exceedance. In this paper we investigate score matching for performing high-dimensional peaks over threshold inference, focusing on extreme value processes associated to log-Gaussian random functions and discuss the behaviour of the proposed estimators for regularly-varying distributions with normalized marginals. Their performance is assessed on grids with several hundred locations, simulating from both the true model and from its domain of attraction. We illustrate the potential and flexibility of our methods by modelling extreme rainfall on a grid with 3600 locations, based on risks for exceedances over local quantiles and for large spatially accumulated rainfall, and briefly discuss diagnostics of model fit. The differences between the two fitted models highlight the importance of the choice of risk and its impact on the dependence structure. * raphael.de-fondeville@epfl.ch † anthony.davison@epfl.ch 1 arXiv:1605.08558v2 [stat.ME]

The distribution of spatially aggregated data from a stochastic process X may exhibit a different tail behavior than its marginal distributions. For a large class of aggregating functionals we introduce the -extremal coefficient that quantifies this difference as a function of the extremal spatial dependence in X. We also obtain the joint extremal dependence for multiple aggregation functionals applied to the same process. Explicit formulas for the -extremal coefficients and multivariate dependence structures are derived in important special cases. The results provide a theoretical link between the extremal distribution of the aggregated data and the corresponding underlying process, which we exploit to develop a method for statistical downscaling. We apply our framework to downscale daily temperature maxima in the south of France from a gridded data set and use our model to generate high resolution maps of the warmest day during the 2003 heatwave.

Peaks-over-threshold analysis using the generalized Pareto distribution is widely applied in modelling tails of univariate random variables, but much information may be lost when complex extreme events are studied using univariate results. In this paper, we extend peaks-over-threshold analysis to extremes of functional data. Threshold exceedances defined using a functional r are modelled by the generalized r-Pareto process, a functional generalization of the generalized Pareto distribution that covers the three classical regimes for the decay of tail probabilities. This process is the only possible limit for the distribution of r-exceedances of a properly rescaled process. We give construction rules, simulation algorithms and inference procedures for generalized r-Pareto processes, discuss model validation, and use the new methodology to study extreme European windstorms and heavy spatial rainfall.

Peaks‐over‐threshold analysis using the generalised Pareto distribution is widely applied in modelling tails of univariate random variables, but much information may be lost when complex extreme events are studied using univariate results. In this paper, we extend peaks‐over‐threshold analysis to extremes of functional data. Threshold exceedances defined using a functional r are modelled by the generalised r‐Pareto process, a functional generalisation of the generalised Pareto distribution that covers the three classical regimes for the decay of tail probabilities, and that is the only possible continuous limit for r‐exceedances of a properly rescaled process. We give construction rules, simulation algorithms and inference procedures for generalised r‐Pareto processes, discuss model validation and apply the new methodology to extreme European windstorms and heavy spatial rainfall.