Bayesian mixture models for extremes

This thesis proposes novel methods to analyse and model extremely high values stemming from multiple data-generating processes. Extreme value theory deals with mathematically modelling the behaviour of the tails of a probability distribution. In many real-world scenarios, observed maximum values deviate from a single parametric distribution, challenging the conventional assumption of being realisations of independent and identically distributed random variables. Examples arise in various domains, such as hydrology (e.g., floods resulting from rainfall and snow-melt, precipitation maxima connected to different weather regimes) and finance (e.g., stock prices bursts linked to the impact of different economic cycles, maximum losses observed in bull and bear market conditions). Incorrectly assuming a single component for the right tail when extremes are actually organised into multiple groups can lead to model misspecification and inaccurate risk estimation for rare events based on return levels. This dissertation focuses on the context of block maxima, in which extreme observations are maxima of non-overlapping blocks. Novel methodologies based on mixture models of distributions for extremes are exploited in a Bayesian framework to capture heterogeneity in the right tail. A first contribution is the development of a twocomponent mixture model of Gumbel distributions, in which the allocation is based on a set of relevant variables, not just the information on the underlying process, which is typically unknown or not useful in separating the right tail. A second contribution is the use of an infinite mixture model of generalised extreme value (GEV) distributions, which captures the grouped structure in the right tail by assigning a Dirichlet process prior to the mixing measure. This approach offers a highly flexible method capable of characterising a wide range of scenarios, without imposing restrictive assumptions on the number of mixture components in the data. The proposed methods are illustrated using both simulated and real-world data, with applications to precipitation.