Extremile Regression

Regression extremiles define a least squares analogue of regression quantiles. They are determined by weighted expectations rather than tail probabilities. Of special interest is their intuitive meaning in terms of expected minima and maxima. Their use appears naturally in risk management where, in contrast to quantiles, they fulfill the coherency axiom and take the severity of tail losses into account. In addition, they are comonotonically additive and belong to both the families of spectral risk measures and concave distortion risk measures. This article provides the first detailed study exploring implications of the extremile terminology in a general setting of presence of covariates. We rely on local linear (least squares) check function minimization for estimating conditional extremiles and deriving the asymptotic normality of their estimators. We also extend extremile regression far into the tails of heavy-tailed distributions. Extrapolated estimators are constructed and their asymptotic theory is developed. Some applications to real data are provided.回归极值（Regression extremiles）是回归分位数的最小二乘类比形式。其定义基于加权期望而非尾部概率，其中基于期望极小值与极大值的直观含义尤为值得关注。在风险管理领域，回归极值的应用自然贴合场景需求：与分位数不同，它满足一致性公理，且可考量尾部损失的严重程度。此外，回归极值具备共单调可加性，同时隶属于谱风险测度与凹失真风险测度两大类别。本文首次针对存在协变量（covariates）的一般情形，详细探讨了极值回归相关术语的应用内涵。研究采用局部线性（最小二乘）检验函数最小化方法，对条件回归极值进行估计，并推导了其估计量的渐近正态性（asymptotic normality）。此外，本文还将极值回归分析拓展至厚尾分布的极端尾部区域，构建了外推估计量并完善了其渐近理论体系。文末提供了基于真实数据的应用实例。