Modelling and Inference for Extremal Dependence in Space and Time

The statistical modelling of spatial extremes in an important tool to quantify and aggregate the risk of joint occurrences of extreme events at different locations in space. The classical spatial models for extreme data, such as max-stable processes, are supported by a solid asymptotic theory, but may be too rigid to describe the behaviour of data at finite, sub-asymptotic levels. For this reason, models with a more flexible extremal dependence structure have been proposed. One important class is that of random scale mixtures, defined as a product between a random variable and a spatial stochastic process. Two special cases in this class are Gaussian scale mixtures and Gaussian location mixtures. In this thesis, a more general class including these two cases is formalized, and new models in this class are proposed and studied. Moreover, a new efficient inferential method for models in this class is developed. Then, another class of models that extends random scale mixtures to a spatio-temporal framework is proposed. The new models allow to obtain different extremal dependence for pairs in space and time. A simulation-based estimation method is proposed for these models, due to the intractability of their likelihood functions. The models presented in the thesis are applied in two real data examples about the spatial dependence of wildfire risk and the spatio-temporal dependence of extreme rainfall.