Bayesian analysis of extreme events with threshold estimation

The aim of this paper is to analyse extremal events using generalized Pareto distributions (GPD), considering explicitly the uncertainty about the threshold. Current practice empirically determines this quantity and proceeds by estimating the GPD parameters on the basis of data beyond it, discarding all the information available below the threshold. We introduce a mixture model that combines a parametric form for the center and a GPD for the tail of the distributions and uses all observations for inference about the unknown parameters from both distributions, the threshold included. Prior distributions for the parameters are indirectly obtained through experts quantiles elicitation. Posterior inference is available through Markov chain Monte Carlo methods. Simulations are carried out in order to analyse the performance of our proposed model under a wide range of scenarios. Those scenarios approximate realistic situations found in the literature. We also apply the proposed model to a real dataset, Nasdaq 100, an index of the financial market that presents many extreme events. Important issues such as predictive analysis and model selection are considered along with possible modeling extensions.