Resumen de Estimación bayesiana de cópulas extremales en procesos de Poisson
María Isabel Ortego Martínez 
The estimation of occurrence probabilities of extremal quantities is essential in the study of hazards associated with natural phenomena. The extremal quantities of interest usually correspond to phenomena characterized by two or more magnitudes, often showing dependence among them. In order to better characterize situations that could be dangerous, the magnitudes that describe the phenomenon should be jointly described. A Poisson-GPD model, which describes the occurrence of extremal events and their marginal sizes, has been established: the occurrence of the extremal events is represented by means of a Poisson process, and each event is characterized by a size modelled by a Generalized Pareto Distribution, GPD. The dependence between events is modelled through copula functions: a family of Gumbel copulas, suitable for the type of data treated, and a new type of copula that is introduced, the CrEnC copula. The CrEnC copula minimizes the mutual information in situations in which only partial information in the form of restrictions is available, such as marginal models or joint moments of the variables. In this context, data are often scarce, and the uncertainty in the estimation of the model will be great. A Bayesian estimation process that takes into account this uncertainty has been established. Goodness-of-fit of some aspects of the model (GPD goodness-of-fit, GPD-Weibull hypothesis and global goodness-of-fit) has been checked using a selection of Bayesian p-values, which incorporate the uncertainty of the parameter estimation. Once the model has been estimated, a post-process of information has been performed to obtain a posteriori quantities of interest, such as exceedance probabilities of reference values or return periods of events of a certain size. The proposed model is applied to three datasets, with different characteristics. The results obtained are good: the introduced CrEnC copulas correctly represent the dependence in situations in which only partial information is available, and the Bayesian estimation of the parameters of the model gives added value to the results, because it allows the uncertainty of the posterior estimates, such as hazard and dependence parameters, to be evaluated.
