Practical strategies for regression models for extremes
This article contributes to a widely overlooked issue related to the restrictions inherited by finite-sample maxima distributions when they are approximated by the generalised extreme value (GEV) distribution. Indeed, 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 when the shape parameter is positive, are inherited by the finite-sample maxima, which might not have bounded support. This is particularly problematic in a regression setting, where the lower bound also depends on covariates.
To tackle this issue, we construct an alternative distribution for block-maxima based on the GEV distribution but with unbounded support, where the artificial lower bound imposed by the GEV approximation is avoided.
As a second contribution, we propose a new parametrisation of the GEV distribution that differs from the usual location-scale parametrisation. Our reparametrisation has a meaningful interpretation even when the first two moments of the GEV distribution are not defined. Moreover, it has natural connections with empirical quantiles, and it is particularly advantageous in the Bayesian framework as prior distributions can be easily assigned.
As our third and final contribution, we introduce the concept of property-preserving penalised complexity (P3C) priors to preserve important model properties when these properties are not “continuous” as functions of model parameters. We use this approach to avoid inconsistencies related to the existence of first and second moments.
We illustrate our methods with an application to NO2 pollution levels in California, which reveals the robustness of our proposed distribution, as well as the suitability of the new parametrisation and the P3C prior framework.