Extremile (Daouia, Gijbels and Stupfler,2019) is a novel and coherent measure of risk, determined by weighted expectations rather than tail probabilities. It finds application in risk management, and, in contrast to quantiles, it fulfills the axioms of consistency, taking into account the severity of tail losses. However, existing studies (Daouia, Gijbels and Stupfler,2019,2022) on extremile involve unknown distribution functions, making it challenging to obtain a root n-consistent estimator for unknown parameters in linear extremile regression. This article introduces a new definition of linear extremile regression and its estimation method, where the estimator is root n-consistent. Additionally, while the analysis of unlabeled data for extremes presents a significant challenge and is currently a topic of great interest in machine learning for various classification problems, we have developed a semi-supervised framework for the proposed extremile regression using unlabeled data. This framework can also enhance estimation accuracy under model misspecification. Both simulations and real data analyses have been conducted to illustrate the finite sample performance of the proposed methods.

翻译：极值分位数（Daouia, Gijbels and Stupfler,2019）是一种新颖且一致的风险度量指标，它由加权期望而非尾部概率决定。该指标在风险管理中具有应用价值，与分位数不同，它满足一致性公理，能够考虑尾部损失的严重性。然而，现有关于极值分位数的研究（Daouia, Gijbels and Stupfler,2019,2022）涉及未知分布函数，使得在线性极值分位数回归中难以获得未知参数的根n一致估计量。本文提出了一种线性极值分位数回归的新定义及其估计方法，所得到的估计量具有根n一致性。此外，针对极端值的无标签数据分析构成重大挑战，且当前是机器学习中各类分类问题备受关注的研究热点，我们基于无标签数据为所提出的极值分位数回归开发了一个半监督框架。该框架还能在模型误设情况下提高估计精度。通过模拟实验和实际数据分析，验证了所提方法在有限样本下的表现。