Threshold-based approaches in geostatistics of extremes

Abstract

Extreme Value Theory (EVT) is an appropriate tool for the stochastic analysis of risks related to extreme outcomes in random processes. If such processes are spatial in extent, as for example precipitation, wind or storm tides and surges, we have to deal with spatial variability and dependence across a high-dimensional sample space. Multivariate EVT deals with limit distributions for linearly normalized observations which are max-stable if we use maxima and of Generalized Pareto type if we use exceedances over a high threshold. More generally, tail behavior can be characterized by (multivariate) regular variation. In all of these cases, the multivariate dependence structure can be characterized by the so-called angular measure for which threshold-based estimators have been proposed. In the infinite-dimensional setting, we use the framework of max-stable limit processes. Based upon weak convergence, it allows to generalize multivariate EVT and its statistical tools. In this talk, we generalize the spectral measure approach to spatial extremes, yielding the spectrogram as a new diagnostic and inferential tool. We illustrate its utility by visualizing theoretical spectrograms for common max-stable models and empirical spectrograms for real and simulated data. In a first part, we give an overview of univariate and multivariate EVT, with a focus on the angular measure and its estimation. Then we lead over to spatial max-stable models, recall the principal inferential methods and present the new results. Major differences to classical geostatistics are mentioned in passing. We conclude with an outlook on current “hot topics” in geostatistics of extremes. 1