Bayesian Inference for Peaks Over Threshold Models for Multivariate and Spatial Extremes

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Extreme value theory is a branch of probability and statistics that focuses on the study of rare events. There are many areas of science and technology where such methods find applications. Examples include the quantification of actuarial risk, estimation of large fluctuations in financial markets, and the estimation of maximum water flow. Of particular relevance for our society is the study of extreme climate events. Historical records of climate related variables provide evidence that there is an intensification of extreme weather. Climate projections indicate that the frequency and intensity of events with catastrophic potential will increase even further. This research focuses on the development of statistical methods that will enable careful assessment of the uncertainties related to extreme events. The proposed methods will focus on models that look jointly at several variables and apply to observations collected in large spatial domains. Probabilistic assessment of the uncertainties in the occurrence of rare events will be made possible by a Bayesian approach. This will provide a powerful tool for rational decision and policy making.In this project, novel methodology for the statistical analysis of the distributions of extreme values is proposed. The methods are based on using the amounts in excess of a fixed threshold for the variables of interest, or peaks over thresholds (POT). POT methods to perform Bayesian inference for (a) multivariate observations, (b) spatially indexed fields, and (c) fields of multivariate observations in space will be developed and implemented. In extreme value theory, the focus is on extrapolation as scarce extreme observations are used to describe the behavior of the tails of the distribution. The theory and the methods for inference on univariate extreme values are firmly established and fully developed. For multivariate problems, it is key to model the joint tail dependence of the different variables. In this sense, the theory is well understood, but inferential methods are not as straightforward as in the univariate case. This is especially true for POT methods. A further level of complication is introduced when dealing with georeferenced data. In fact, in the spatial setting, it is impossible to write the full likelihood of realistic POT models for observations collected at an arbitrary number of locations. This research focuses on the development of methods that (a) are conceptually clear to specify using a simple factorization that is at the core of Bayesian hierarchical models, (b) allow for fully integrated Bayesian inference that accounts for all estimation uncertainty and quantifies it probabilistically, (c) have theoretically sound asymptotic properties, (d) provide flexible characterizations of a wide range of tail dependence, and (e) are computationally feasible for large spatial domains.