Functional Peaks-Over-Threshold Analysis for Complex Extreme Events

Most current risk assessment for complex extreme events relies on catalogues of similar events, either historical or generated artificially. In the latter, no existing methods produce completely new events with mathematically justified extrapolation above observed level of severity. This thesis contributes to the development of stochastic generators of events based on extreme value theory, with a special focus on natural hazards.

The sources of historical meteorological records are multiple but climate model output is attractive for its spatial completeness and homogeneity. From a statistical perspective, these are massive gridded data sets, which can be exploited for accurate estimation of extreme events. The first contribution of this thesis describes methods of statistical inference for extremal processes that are computationally tractable for large data sets. We also relate the extremal behaviour of aggregated data to point observations, a result that we use to downscale gridded data to local tail distributions. These contributions are illustrated by applications to rainfall and heatwaves.

Building stochastic generators of extreme events requires the extension of classical peaks-over-threshold analysis to continuous stochastic processes. We develop a framework in which characterization of complex extremes can be motivated by field-specific expertise. The contribution includes the description of the theoretical limiting distribution of functional exceedances, called the generalized r-Pareto process, the functional equivalent of the generalized Pareto distribution, for which we describe statistical inference procedures, simulation algorithms and goodness-of-fit diagnostics. We apply these results to build a stochastic weather generator of extreme wind storms over Europe.