Recently defined expectile regions capture the idea of centrality with respect to a multivariate distribution, but fail to describe the tail behavior while it is not at all clear what should be understood by a tail of a multivariate distribution. Therefore, cone expectile sets are introduced which take into account a vector preorder for the multi-dimensional data points. This provides a way of describing and clustering a multivariate distribution/data cloud with respect to an order relation. Fundamental properties of cone expectiles including dual representations of both expectile regions and cone expectile sets are established. It is shown that set-valued sublinear risk measures can be constructed from cone expectile sets in the same way as in the univariate case. Inverse functions of cone expectiles are defined which should be considered as rank functions rather than depth functions. Finally, expectile orders for random vectors are introduced and characterized via expectile rank functions.

翻译：最近定义的期望分位数区域捕捉了关于多元分布的中心性概念，但未能描述尾部行为，且多元分布"尾部"的含义尚不明确。为此，引入了锥期望分位数集，该集合考虑了多维数据点的向量预序关系，从而提供了一种基于序关系对多元分布/数据云进行描述与聚类的方法。建立了锥期望分位数的基本性质，包括期望分位数区域与锥期望分位数集的对偶表示。研究表明，与单变量情形类似，可通过锥期望分位数集构造集值次线性风险测度。定义了锥期望分位数的逆函数，该函数应被视为秩函数而非深度函数。最后，引入了随机向量的期望分位数序，并通过期望分位数秩函数对其进行了刻画。