This paper focuses on semi-parametric estimation of multivariate expectiles for extreme levels of risk. Multivariate expectiles and their extremes have been the focus of plentiful research in recent years. In particular, it has been noted that due to the difficulty in estimating these values for elevated levels of risk, an alternative formulation of the underlying optimization problem would be necessary. However, in such a scenario, estimators have only been provided for the limiting cases of tail dependence: independence and comonotonicity. In this paper, we extend the estimation of multivariate extreme expectiles (MEEs) by providing a consistent estimation scheme for random vectors with any arbitrary dependence structure. Specifically, we show that if the upper tail dependence function, tail index, and tail ratio can be consistently estimated, then one would be able to accurately estimate MEEs. The finite-sample performance of this methodology is illustrated using both simulated and real data.

本文着重于针对极端风险水平的多元期望的半参数估计。近年来，多变量期望值及其极端值已成为众多研究的焦点。特别是，已经注意到，由于难以估计这些值以提高风险水平，因此有必要对潜在的优化问题进行替代性表述。但是，在这种情况下，仅针对尾部依赖的有限情况（独立性和共调性）提供了估计量。在本文中，我们通过为具有任意依赖性结构的随机向量提供一致的估计方案，扩展了多变量极端期望（MEE）的估计。具体而言，我们表明，如果可以一致地估计上尾巴依赖函数，尾巴索引和尾巴比率，这样便可以准确估计MEE。使用模拟数据和实际数据说明了该方法的有限样本性能。