Practical strategies for generalized extreme value-based regression models for extremes

The generalized extreme value (GEV) distribution is the only possible limiting distribution of properly normalized maxima of a sequence of independent and identically distributed random variables. As such, it has been widely applied to approximate the distribution of maxima over blocks. In these applications, GEV properties such as finite lower endpoint when the shape parameter xi$$ \xi $$ is positive or the loss of moments due to the magnitude of xi$$ \xi $$ are inherited by the finite-sample maxima distribution. The extent to which these properties are realistic for the data at hand has been widely ignored. Motivated by these overlooked consequences in a regression setting, we here make three contributions. First, we propose a blended GEV (bGEV) distribution, which smoothly combines the left tail of a Gumbel distribution (GEV with xi=0$$ \xi =0 $$) with the right tail of a Frechet distribution (GEV with xi>0$$ \xi >0 $$). Our resulting distribution has, therefore, unbounded support. Second, we proposed a principled method called property-preserving penalized complexity (P3$$ {}<^>3 $$C) prior to decide on the existence of the GEV distribution first and second moments a priori. Third, we propose a reparametrization of the GEV distribution that provides a more natural interpretation of the (possibly covariate-dependent) model parameters, which in turn helps define meaningful priors. We implement the bGEV distribution with the new parameterization and the P3$$ {}<^>3 $$C prior approach in the R-INLA package to make it readily available to users. We illustrate our methods with a simulation study that reveals that the GEV and bGEV distributions are comparable when estimating the right tail under large-sample settings. Moreover, some small-sample settings show that the bGEV fit slightly outperforms the GEV fit. Finally, we conclude with an application to NO2$$ {}_2 $$ pollution levels in California that illustrates the suitability of the new parameterization and the P3$$ {}<^>3 $$C prior approach in the Bayesian framework.

Automated summary

The generalized extreme value (GEV) distribution is widely used to approximate the distribution of maxima over blocks. However, the realistic properties of GEV distribution are often ignored. In this study, we propose a blended GEV (bGEV) distribution, a principled method called property-preserving penalized complexity (P3C) prior, and a reparametrization approach to improve the GEV distribution. Through simulation study and application to NO2 pollution levels in California, we demonstrate that the bGEV distribution performs slightly better than the GEV distribution in certain situations.