Exploiting Structure of Maximum Likelihood Estimators for Extreme Value Threshold Selection

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 to produce complementary plots with greater interpretability, and suggest a simple likelihood-based procedure that allows for automated threshold selection. Supplementary materials for this article are available online.