Automated Threshold Selection and Associated Inference Uncertainty for Univariate Extremes

Threshold selection is a fundamental problem in any threshold-based extreme value analysis. While models are asymptotically motivated, selecting an appropriate threshold for finite samples is difficult and highly subjective through standard methods. Inference for high quantiles can also be highly sensitive to the choice of threshold. Too low a threshold choice leads to bias in the fit of the extreme value model, while too high a choice leads to unnecessary additional uncertainty in the estimation of model parameters. We develop a novel methodology for automated threshold selection that directly tackles this bias-variance tradeoff. We also develop a method to account for the uncertainty in the threshold estimation and propagate this uncertainty through to high quantile inference. Through a simulation study, we demonstrate the effectiveness of our method for threshold selection and subsequent extreme quantile estimation, relative to the leading existing methods, and show how the method’s effectiveness is not sensitive to the tuning parameters. We apply our method to the well-known, troublesome example of the River Nidd dataset.

基于阈值的极值分析中，阈值选取是一项基础性问题。尽管现有极值模型多基于渐近理论推导，但针对有限样本选取合适阈值时，传统方法不仅难度较大，且主观性极强。高阶分位数的统计推断同样对阈值选取极为敏感：阈值选取过低会导致极值模型拟合产生偏倚，而阈值过高则会为模型参数估计引入不必要的额外不确定性。我们提出一种全新的自动化阈值选取方法，可直接解决这一偏倚-方差权衡问题。同时，我们还构建了可量化阈值估计不确定性，并将该不确定性传递至高阶分位数推断过程的方法。通过仿真研究，相较于当前主流的现有方法，我们验证了所提方法在阈值选取及后续极值分位数估计任务中的有效性，并证明该方法的性能对调参参数并不敏感。我们将所提方法应用于广为人知且颇具挑战性的尼德河（River Nidd）数据集案例中。