Extreme Value Statistics in Semi-Supervised Models

We consider extreme value analysis in a semi-supervised setting, where we observe, next to the n data on the target variable, n + m data on one or more covariates. This is called the semi-supervised model with n labeled and m 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 n 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.

本研究在半监督学习环境中探讨极端值分析，其中，在观察目标变量n个数据点的同时，我们还观察了关于一个或多个协变量n+m个数据点。这种半监督模型被称为n个标记数据和m个未标记数据的半监督模型。通过利用目标变量与协变量之间的尾部依赖性，本研究在该环境中推导出目标变量的极端值指数和极端分位数估计量，并建立了它们的渐近行为。与仅基于n个目标变量数据的单变量估计量相比，我们的估计量在渐近方差方面显著改进，而渐近偏差保持不变。一项仿真研究证实了这两种估计量行为的大幅改进。最后，该方法被应用于法国的降雨数据。本文的补充材料可在网上获取，包括用于重现工作的标准化材料描述。