Expectile depth: Theory and computation for bivariate datasets
本文提出双变量期望分位数区域及其深度函数，给出计算极值点和点深度的算法，并证明样本区域的收敛性和一致性，适用于多元数据中的中心性度量。
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 α 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<α≤1∕2 is built up by all points whose univariate projections lie between the expectiles of levels α and 1−α 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.