Bayesian threshold selection for extremal models using measures of surprise

Statistical extreme value theory is concerned with the use of asymptotically motivated models to describe the extreme values of a process. A number of commonly used models are valid for observed data that exceed some high threshold. However, in practice a suitable threshold is unknown and must be determined for each analysis. While there are many threshold selection methods for univariate extremes, there are relatively few that can be applied in the multivariate setting. In addition, there are only a few Bayesian-based methods, which are naturally attractive in the modelling of extremes due to data scarcity. The use of Bayesian measures of surprise to determine suitable thresholds for extreme value models is proposed. Such measures quantify the level of support for the proposed extremal model and threshold, without the need to specify any model alternatives. This approach is easily implemented for both univariate and multivariate extremes.Comment: To appear in Computational Statistics and Data Analysi

Exploiting structure of maximum likelihood estimators for extreme value threshold selection

In order to model the tail of a distribution, one has to define the threshold above or below which an extreme value model produces a suitable fit. Parameter stability plots, whereby one plots maximum likelihood estimates of supposedly threshold-independent parameters against threshold, form one of the main tools for threshold selection by practitioners, principally due to their simplicity. However, one repeated criticism of these plots is their lack of interpretability, with pointwise confidence intervals being strongly dependent across the range of thresholds. In this article, we exploit the independent-increments structure of maximum likelihood estimators in order to produce complementary plots with greater interpretability, and a suggest simple likelihood-based procedure which allows for automated threshold selection