Expectiles induce a law-invariant risk measure that has recently gained popularity in actuarial and financial risk management applications. Unlike quantiles or the quantile-based Expected Shortfall, the expectile risk measure is coherent and elicitable. The estimation of extreme expectiles in the heavy-tailed framework, which is reasonable for extreme financial or actuarial risk management, is not without difficulties; currently available estimators of extreme expectiles are typically biased and hence may show poor finite-sample performance even in fairly large samples. We focus here on the construction of bias-reduced extreme expectile estimators for heavy-tailed distributions. The rationale for our construction hinges on a careful investigation of the asymptotic proportionality relationship between extreme expectiles and their quantile counterparts, as well as of the extrapolation formula motivated by the heavy-tailed context. We accurately quantify and estimate the bias incurred by the use of these relationships when constructing extreme expectile estimators. This motivates the introduction of classes of bias-reduced estimators whose asymptotic properties are rigorously shown, and whose finite-sample properties are assessed on a simulation study and three samples of real data from economics, insurance and finance.

Expectiles 引发了一种法律不变的风险度量，该度量最近在精算和金融风险管理应用中得到普及。与分位数或基于分位数的预期短缺不同，预期风险度量是连贯且可引出的。对于极端财务或精算风险管理而言，重尾框架下的极端期望估计并非没有困难；当前可用的极端期望估计器通常是有偏差的，因此即使在相当大的样本中也可能表现出较差的有限样本性能。我们在这里专注于为重尾分布构建减少偏差的极端期望估计器。我们构建的基本原理取决于对极端期望值与其分位数对应物之间的渐近比例关系的仔细研究，以及由重尾上下文驱动的外推公式。在构建极端期望估计量时，我们准确地量化和估计了使用这些关系所产生的偏差。这促使引入了减少偏差的估计量类别，这些估计量的渐近特性得到严格显示，其有限样本特性通过模拟研究和来自经济、保险和金融的三个真实数据样本进行评估。在构建极端期望估计量时，我们准确地量化和估计了使用这些关系所产生的偏差。这促使引入了减少偏差的估计量类别，这些估计量的渐近特性得到严格显示，其有限样本特性通过模拟研究和来自经济、保险和金融的三个真实数据样本进行评估。在构建极端期望估计量时，我们准确地量化和估计了使用这些关系所产生的偏差。这促使引入了减少偏差的估计量类别，这些估计量的渐近特性得到严格显示，其有限样本特性通过模拟研究和来自经济、保险和金融的三个真实数据样本进行评估。