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The Peaks Over Threshold (POT) method is the most popular statistical methodfor the analysis of univariate extremes. Even though there is a rich appliedliterature on Bayesian inference for the POT method there is no asymptotictheory for such proposals. Even more importantly, the ambitious and challengingproblem of predicting future extreme events according to a proper probabilisticforecasting approach has received no attention to date. In this paper wedevelop the asymptotic theory (consistency, contraction rates, asymptoticnormality and asymptotic coverage of credible intervals) for the Bayesianinference based on the POT method. We extend such an asymptotic theory to coverthe Bayesian inference on the tail properties of the conditional distributionof a response random variable conditionally to a vector of random covariates.With the aim to make accurate predictions of severer extreme events than thoseoccurred in the past, we specify the posterior predictive distribution of afuture unobservable excess variable in the unconditional and conditionalapproach and we prove that is Wasserstein consistent and derive its contractionrates. Simulations show the good performances of the proposed Bayesianinferential methods. The analysis of the change in the frequency of financialcrises over time shows the utility of our methodology.

Asymptotic theory for Bayesian inference and prediction: from the ordinary to a conditional Peaks-Over-Threshold method