  In many applied fields it is desired to make predictions with the aim of
assessing the plausibility of more severe events than those already recorded to
safeguard against calamities that have not yet occurred. This problem can be
analysed using extreme value theory. We consider the popular peaks over a
threshold method and show that the generalised Pareto approximation of the true
predictive densities of both a future unobservable excess or peak random
variable can be very accurate. We propose both a frequentist and a Bayesian
approach for the estimation of such predictive densities. We show the
asymptotic accuracy of the corresponding estimators and, more importantly,
prove that the resulting predictive inference is asymptotically reliable. We
show the utility of the proposed predictive tools analysing extreme
temperatures in Milan in Italy.
