Multivariate expectile trimming and the BExPlot

Expectiles are the solution to an asymmetric least squares minimization problem for univariate data. They resemble some similarities with the quantiles, and just like them, expectiles are indexed by a level α. In the present paper, we introduce and discuss the main properties of the expectile multivariate trimmed regions, a nested family of sets, whose instance with trimming level α is built up by all points whose univariate projections lie between the expectiles of levels α and 1 − α of the projected dataset. Such trimming 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 study here the convergence of the sample expectile trimmed regions to the population ones and the uniform consistency of the sample expectile depth. We also provide efficient algorithms for determining the extreme points of the expectile regions as well as for computing the depth of a point in R2. These routines are based on circular sequence constructions. Finally, we present some real data examples for which the Bivariate Expectile Plot (BExPlot) is introduced.