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似方法。其通过加权期望而非尾部概率加以定义。其中，其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，回归极值的应用天然适配场景需求：与分位数不同，它满足一致性公理，且能够充分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，同时属于谱风险测度与凹扭曲风险测度两大范畴。

本文首次在存在协变量的一般框架下，对极值术语的相关内涵展开详细且系统的研究。我们采用局部线性（最小二乘）检验函数最小化方法，对条件极值进行估计，并推导其估计量的渐近正态性。此外，我们将极值回归拓展至厚尾分布的极端尾部区间，构建了外推估计量并建立其渐近理论。最后，本文提供了若干实际数据的应用案例。