Practical strategies for GEV-based regression models for extremes

Abstract

The generalised extreme value (GEV) distribution is a three parameter family that describes the asymptotic behaviour of properly renormalised maxima of a sequence of independent and identically distributed random variables. If the shape parameter $\xi$ is zero, the GEV distribution has unbounded support, whereas if $\xi$ is positive, the limiting distribution is heavy-tailed with infinite upper endpoint but finite lower endpoint. In practical applications, we assume that the GEV family is a reasonable approximation for the distribution of maxima over blocks, and we fit it accordingly. This implies that GEV properties, such as finite lower endpoint in the case $\xi>0$, are inherited by the finite-sample maxima, which might not have bounded support. This is particularly problematic when predicting extreme observations based on multiple and interacting covariates. To tackle this usually overlooked issue, we propose a blended GEV distribution, which smoothly combines the left tail of a Gumbel distribution (GEV with $\xi=0$) with the right tail of a Fr\'echet distribution (GEV with $\xi>0$) and, therefore, has unbounded support. Using a Bayesian framework, we reparametrise the GEV distribution to offer a more natural interpretation of the (possibly covariate-dependent) model parameters. Independent priors over the new location and spread parameters induce a joint prior distribution for the original location and scale parameters. We introduce the concept of property-preserving penalised complexity (P$^3$C) priors and apply it to the shape parameter to preserve first and second moments. We illustrate our methods with an application to NO$_2$ pollution levels in California, which reveals the robustness of the bGEV distribution, as well as the suitability of the new parametrisation and the P$^3$C prior framework.