Extreme Value Statistics in Semi-Supervised Models

We consider extreme value analysis in a semi-supervised setting, where we observe, next to the <i>n</i> data on the target variable, <i>n</i> + <i>m</i> data on one or more covariates. This is called the semi-supervised model with <i>n</i> labeled and <i>m</i> unlabeled data. By exploiting the tail dependence between the target variable and the covariates, we derive estimators for the extreme value index and extreme quantiles of the target variable in this setting and establish their asymptotic behavior. Our estimators substantially improve the univariate estimators, based on only the <i>n</i> target variable data, in terms of asymptotic variances whereas the asymptotic biases remain unchanged. A simulation study confirms the substantially improved behavior of both estimators. Finally the estimation method is applied to rainfall data in France. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

本文针对半监督场景（semi-supervised setting）下的极值分析（extreme value analysis）问题展开研究：即在观测到n条目标变量（target variable）数据的同时，还可获得n+m条来自一个或多个协变量（covariates）的数据，此类场景下的模型即为包含n条标注数据（labeled data）与m条未标注数据（unlabeled data）的半监督模型。通过利用目标变量与协变量间的尾部相依性（tail dependence），本文推导得到该场景下目标变量的极值指数（extreme value index）与极端分位数（extreme quantiles）估计量，并论证了其渐近性质（asymptotic behavior）。相较于仅基于n条目标变量数据的单变量估计量（univariate estimators），本文所提估计量在渐近方差（asymptotic variances）层面实现了显著优化，且渐近偏差（asymptotic biases）保持不变。仿真研究（simulation study）验证了两类估计量均能实现性能的显著提升。最后，本文将所提估计方法应用于法国降雨量数据集。本文的补充材料可在线获取，其中包含可用于复现研究工作的标准化材料说明。