  Predictions of the uncertainty associated with extreme events are a vital
component of any prediction system for such events. Consequently, the
prediction system ought to be probabilistic in nature, with the predictions
taking the form of probability distributions. This paper concerns probabilistic
prediction systems where the data is assumed to follow either a generalized
extreme value distribution (GEV) or a generalized Pareto distribution (GPD). In
this setting, the properties of proper scoring rules which facilitate the
assessment of the prediction uncertainty are investigated and closed-from
expressions for the continuous ranked probability score (CRPS) are provided. In
an application to peak wind prediction, the predictive performance of a GEV
model under maximum likelihood estimation, optimum score estimation with the
CRPS, and a Bayesian framework are compared. The Bayesian inference yields the
highest overall prediction skill and is shown to be a valuable tool for
covariate selection, while the predictions obtained under optimum CRPS
estimation are the sharpest and give the best performance for high thresholds
and quantiles.
