Expectiles are the solution to an asymmetric least squares minimization problem for univariate data. They resemble the quantiles, and just like them, expectiles are indexed by a level $\alpha$ in the unit interval. In the present paper, we introduce and discuss the main properties of the (multivariate) expectile regions, a nested family of sets, whose instance with level $0<\alpha \le 1∕2$ is built up by all points whose univariate projections lie between the expectiles of levels $\alpha$ and $1-\alpha$ of the projected dataset. Such level is interpreted as the degree of centrality of a point with respect to a multivariate distribution and therefore serves as a depth function. We propose here algorithms for determining all the extreme points of the bivariate expectile regions as well as for computing the depth of a point in the plane. We also study the convergence of the sample expectile regions to the population ones and the uniform consistency of the sample expectile depth. Finally, we present some real data examples for which the Bivariate Expectile Plot (BExPlot) is introduced.

期望值是单变量数据的不对称最小二乘最小化问题的解决方案。它们类似于分位数，就像它们一样，按级别对索引进行索引$\alpha$在单位间隔内。在本文中，我们介绍并讨论了（多变量）期望区域的主要属性，即嵌套集的族，其实例与级别$0<\alpha \le \mathrm{1个}∕\mathrm{2个}$ 由所有变量组成，这些变量的单变量投影位于水平的期望值之间 $\alpha$ 和 $\mathrm{1个}-\alpha$投影数据集。该级别被解释为点相对于多元分布的中心程度，因此用作深度函数。我们在这里提出用于确定双变量期望区域的所有极点以及用于计算平面中某个点的深度的算法。我们还研究了样本预期区域对总体区域的收敛性以及样本预期深度的均匀一致性。最后，我们介绍了一些实际数据示例，为此引入了双变量期望图（BExPlot）。