A1148
Title: Bayesian inference for functional extreme events defined via partially unobserved processes
Authors: Max Thannheimer - University of Stuttgart (Germany) [presenting]
Marco Oesting - University of Stuttgart (Germany)
Abstract: In order to describe the extremal behaviour of some stochastic process $X$, approaches from univariate extreme value theory are typically generalized to the spatial domain. A generalized peaks-over-threshold approach can be used, allowing the consideration of single extreme events. These can be flexibly defined as exceedances of a risk functional $\ell$, such as a spatial average, applied to $X$. Inference for the resulting limit process, the so-called $\ell$-Pareto process, requires the evaluation of $\ell(X)$ and thus the knowledge of the whole process $X$. In practical application, the challenge is that observations of $X$ are only available at single sites. To overcome this issue, a two-step MCMC algorithm is proposed in a Bayesian framework. First, it is sampled from $X$ conditionally on the observations to evaluate which observations lead to $\ell$-exceedances. In the second step, these exceedances are used to sample from the posterior distribution of the parameters of the limiting $\ell$-Pareto process. Alternating these steps results in a full Bayesian model for the extremes of $X$. It is shown that, under appropriate assumptions, the probability of classifying an observation as $\ell$-exceedance in the first step converges to the desired probability. Furthermore, given the first step, the distribution of the Markov chain constructed in the second step converges to the posterior distribution of interest.
