On statistical risk assessment for spatial processes.

Summary Probabilistic modeling of climate and environmental events must take into account their spatial nature. This thesis focuses on the study of risk measures for spatial processes. In a first part, we introduce risk measures able to take into account the dependency structure of the underlying spatial processes when dealing with environmental data. A second part is devoted to the estimation of parameters of max-mix processes. The first part of the thesis is dedicated to risk measures. We extend the work done in [44] on the one hand to Gaussian processes, on the other hand to other max-stable processes and to max-mix processes, other dependence structures are thus considered. The risk measures considered are based on the average L(A,D) of losses or damages D over a region of interest A. We then consider the expectation and variance of these normalized damages. First, we focus on the axiomatic properties of the risk measures, their computation and their asymptotic behavior (when the size of the region A tends to infinity). We compute the risk measures in different cases. For a Gaussian process, X, we consider the excess function: D+ X,u = (X-u)+ where u is a fixed threshold. For max-stable and max-mixed processes X, we consider the power function: DνX = Xν. In some cases, semi-explicit formulas for the corresponding risk measures are given. A simulation study tests the behavior of the risk measures with respect to the many parameters involved and the different forms of the correlation kernel. We also evaluate the computational performance of the different proposed methods. This one is satisfactory. Finally, we have used a previous study on pollution data in the Italian Piedmont, which can be considered as Gaussian. We study the risk measure associated to the legal pollution threshold given by the European directive 2008/50/EC. In a second part, we propose a procedure for estimating the parameters of a max-mix process, as an alternative to the composite maximum likelihood estimation method. This more classical method of estimation by composite maximum likelihood is especially efficient to estimate the parameters of the max-stable part of the mixture (and less efficient to estimate the parameters of the asymptotically independent part). We propose a least squares method based on the F-madogram: minimization of the quadratic difference between the theoretical F-madogram and the empirical F-madogram. This method is evaluated by simulation and compared to the composite maximum likelihood method. The simulations indicate that the F-madogram least squares method performs better in estimating the parameters of the asymptotically independent part.