On statistical inference for extreme spatial and spatio-temporal processes.

Summary Natural disasters, such as heat waves, storms or extreme precipitation, originate from physical processes and have, by nature, a spatial or spatiotemporal dimension. The development of models and inference methods for these processes is a very active research area. This thesis deals with statistical inference for extreme events in the spatial and spatiotemporal framework. In particular, we are interested in two classes of stochastic processes: spatial max-mixing processes and spatio-temporal max-stable processes. We illustrate the results obtained on precipitation data in eastern Australia and in a region of Florida in the United States. In the spatial part, we propose two tests on the mixing parameter a of a spatial max-mixing process: the statistical test Za and the pairwise likelihood ratio LRa. We compare the performances of these tests on simulations. We use the pairwise likelihood for estimation. Overall, the performance of both tests is satisfactory. However, the tests encounter difficulties when the parameter a is on the boundary of the parameter space, i.e., a ∈ {0,1}, due to the presence of "nuisance" parameters that are not identified under the null hypothesis. We apply these tests in the context of an excess analysis over a large threshold for rainfall data in eastern Australia. We also propose a new estimation procedure to fit spatial max-mix processes when the extreme dependence class is not known. The novelty of this procedure is that it allows inference to be made without first specifying the family of distributions, thus letting the data speak for themselves and guide the estimation. In particular, the estimation procedure uses a least squares fit on the Fλ-madogram expression of a max-mix model that contains the parameters of interest. We show the convergence of the estimator of the mixture parameter a. An indication of asymptotic normality is given numerically. A simulation study shows that the proposed method improves the empirical coefficients for the class of max-mix models. We implement our estimation procedure on monthly rainfall maxima data in Australia for exploratory and confirmatory purposes. In the spatio-temporal part, we propose a semi-parametric estimation method for spatio-temporal max-stable processes based on an explicit expression of the spatio-temporal F-madogram. This part bridges the gap between geostatistics and extreme value theory. In particular, for regular grid observations, we estimate the spatio-temporal F-madogram by its empirical version and we apply a moment-based procedure to obtain the estimates of the parameters of interest. We illustrate the performance of this procedure by a study on simulations. Next, we apply this method to quantify the extremal behavior of maximum precipitation radar data in the state of Florida. This method can be an alternative or a first step for the composite likelihood. Indeed, the semi-parametric estimates could be used as a starting point for the optimization algorithms used in the pairwise likelihood method, in order to reduce the computation time but also to improve the efficiency of the method.