Set-valued expectiles for ordered data analysis

Expectile regions–like depth regions in general–capture the idea of centrality of multivariate distributions. If an order relation is present for the values of random vectors and a decision maker is interested in dominant/best points with respect to this order, centrality is not a useful concept. Therefore, cone expectile sets are introduced which depend on a vector preorder generated by a convex cone. This provides a way of describing and clustering a multivariate distribution/data cloud with respect to an order relation. Fundamental properties of cone expectiles are established including dual representations of both expectile regions and cone expectile sets. 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 ranking functions related to the initial order relation rather than as depth functions. Finally, expectile orders for random vectors are introduced and characterized via expectile ranking functions.