12/06/2011 08:35 AM
Applied Mathematics & Statistics
We develop Bayesian nonparametric modelling and inference methods for the analysis of extremes. We use a point process approach under which the pairwise observations, comprising the time of excesses and the exceedances over a high threshold, are assumed to arise from a non-homogeneous Poisson process. To understand and capture the behavior of rare events, we propose a nonparametric Dirichlet process mixture model for the point process intensity. Particular emphasis is placed on the choice of the mixture kernel to ensure desirable results for the implied tail behavior of the marginal extreme value distribution. At the same time, the mixture nature of the nonparametric model for the intensity of extremes enables more general inferences than traditional parametric methods, which capture temporal heterogeneity for the occurrence of extremes. In particular, the modelling framework yields flexible inference for the joint intensity of extremes, for the marginal intensity over time, and for different types of return level curves. The methodology is illustrated with a simulated data example, and with data involving negative log returns of the Dow Jones index over a five year period.