Statistical Prediction of Peaks Over a Threshold
超阈值极端峰值的统计预测AI 解读 ↓
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超阈值极端峰值的统计预测
- 研究问题
- 如何预测尚未观测到的更极端事件，构造尾部预测分布与区间，并将极值预测同风险度量及时间序列分析连接起来？
- 方法与思路
- 作者采用超阈值法与广义帕累托近似，以频率学派和贝叶斯方法估计预测密度，利用阈值稳定性向深尾部延伸并处理线性时间序列。
- 材料陈述的贡献
- 摘要称，作者建立估计及预测推断的渐近有效性，说明尾部等价预测可恢复常用风险度量，并应用于米兰高温和道指负收益数据。
- 阅读关注点
- 应核对阈值选择、尾部近似和尾部等价的条件、参数不确定性如何进入预测区间，以及时间依赖下的推断与风险度量对应关系。
- 阅读边界
- 这是 arXiv 预印本，不能确认同行评审或期刊接收。摘要未列明尾部及依赖条件，也未给出深尾部外推的有限样本误差。
原文摘要
In many applied fields, the prediction of more severe events than those already recorded is crucial for safeguarding against potential future calamities. What-if analyses, which evaluate hypothetical scenarios up to the worst-case event, play a key role in assessing the potential impacts of extreme events and guiding the development of effective safety policies. This problem can be analyzed using extreme value theory. We employ the well-established peaks-over-threshold method and describe a comprehensive toolkit to address forecasting needs. We examine an \lq\lq out-of-sample" variable and focus on its conditional probability of exceeding a high threshold, representing the predictive distribution of future extreme peaks. We demonstrate that the generalized Pareto approximation of the corresponding predictive density can be remarkably accurate. We then introduce frequentist methods and a Bayesian approach for estimating this predictive density, enabling the derivation of informative predictive intervals. By leveraging threshold stability, we illustrate how predictions can be reliably extended deep into the tail of the unknown data distribution. We establish the asymptotic accuracy of the proposed estimators and, more importantly, prove that the resulting predictive inference is asymptotically valid. Forecasters satisfying the tail-equivalence property allow to recover widely used risk measures for risk assessment through point forecasts. This insight lays the groundwork for a new perspective that integrates risk assessment into the statistical predictive toolbox. Finally, we extend the prediction framework to the case of linear time series. We apply the proposed predictive tools to two real-world datasets: summer peak temperatures recorded in Milan, Italy, over the past 30 years, and daily negative log-returns of the Dow Jones Industrial Average observed over 30 years.