Bayesian Semiparametrics for Modelling the Clustering of Extreme Values

Extreme events can be statistically characterised as excesses of a high threshold. Inference in this case has to account for dependence between excesses. The peaks over threshold approach suggests pre-processing the series by defining clusters of successive observations and making inference only on maxima of those clusters, which can be assumed independent. There exists however no good method for specifying the clusters, causing quantiles for long return periods to be potentially badly biased. The peaks over threshold method uses asymptotic results as approximation at finite levels; our goal is to focus on the subasymptotic model suggested by Eastoe and Tawn (2012) to develop new Bayesian semiparametric techniques to properly approach the problem, thus getting an estimation of the full excess distribution uncertainty. A simulation study shows the instability in estimated quantiles based on the peaks over threshold method compared to the subasymptotic model across different cluster definitions. An application on river peakflows is also presented and shows how the subasymptotic model, fitted with the novel semiparametric Bayesian method, can be applied to real data. This work is divided into three main parts: an introductory part which provides an insight into multivariate extremes, with a particular attention to special kinds of dependence involved in this framework. In the second part we introduce a conditional multivariate model and develop a semiparametric Gibbs sampler to fit this particular model. The last part deals with the subasymptotic approach, meant to model short-range dependence of excesses over a threshold. We discuss further improvements and alternatives in the last section.