The Peaks Over Threshold (POT) method is the most popular statistical method for the analysis of univariate extremes. Even though there is a rich applied literature on Bayesian inference for the POT, the asymptotic theory for such proposals is missing. Even more importantly, the ambitious and challenging problem of predicting future extreme events according to a proper predictive statistical approach has received no attention to date. In this paper we fill this gap by developing the asymptotic theory of posterior distributions (consistency, contraction rates, asymptotic normality and asymptotic coverage of credible intervals) and prediction within the Bayesian framework in the POT context. We extend this asymptotic theory to account for cases where the focus is on the tail properties of the conditional distribution of a response variable given a vector of random covariates. To enable accurate predictions of extreme events more severe than those previously observed, we derive the posterior predictive distribution as an estimator of the conditional distribution of an out-of-sample random variable, given that it exceeds a sufficiently high threshold. We establish Wasserstein consistency of the posterior predictive distribution under both the unconditional and covariate-conditional approaches and derive its contraction rates. Simulations show the good performances of the proposed Bayesian inferential methods. The analysis of the change in the frequency of financial crises over time shows the utility of our methodology.

翻译：峰值超阈值（POT）方法是单变量极值分析中最常用的统计方法。尽管关于POT贝叶斯推断的应用文献十分丰富，但此类方法的渐近理论仍属空白。更重要的是，基于严格预测统计方法对未来极端事件进行预测这一具有挑战性的难题，迄今尚未得到关注。本文通过发展POT背景下贝叶斯框架内后验分布（一致性、收缩率、渐近正态性与可信区间渐近覆盖）及预测的渐近理论填补了这一空白。我们将该渐近理论拓展至关注响应变量在随机协变量向量条件下的条件分布尾部特性的情形。为准确预测比历史观测更严重的极端事件，我们推导了后验预测分布作为样本外随机变量在超过充分高阈值条件下的条件分布估计量。我们在无条件与协变量条件两种方法下建立了后验预测分布的Wasserstein一致性，并推导了其收缩率。仿真实验表明所提贝叶斯推断方法具有良好的性能。对金融危机频率随时间变化规律的分析验证了本方法的实用性。