Lower Cone Distribution Functions and Set-Valued Quantiles Form Galois Connections

Set-Valued T-Translative Functions and Their Applications in Finance

“…The upper level set ([37], Prop. 4) and also [38]). For a fixed p ∈ [0, 1], the function X → Q X,C (p) is T-translative for Tz = z1I.…”

“…In this case, as in the given reference, the "nonlinear space" is a complete lattice of sets that carries the algebraic structure of a collinear space (see [13]). Another application of inf-extensions to IR d -valued random variables and their random set extensions can be found in [38].…”

Computation of quantile sets for bivariate data

“…The basic assumption is that there is a preference relation for the two-dimensional data points in form of a vector preorder, i.e., a reflexive and transitive relation which is compatible with the algebraic operations in IR 2 . Such vector preorders are in one-two-one correspondence with convex cones C ⊆ IR 2 including 0 ∈ IR 2 via y ≤ C z ⇔ z − y ∈ C…”

“…Moreover, it was shown in [2] that set-valued quantiles and the lower cone distribution functions form Galois connections between complete lattices of sets and the interval [0,1] of real number. This generalizes a property which is straightforward and well-known in the univariate case, but has never been discussed with respect to depth functions and depth regions.…”

“…This generalizes a property which is straightforward and well-known in the univariate case, but has never been discussed with respect to depth functions and depth regions. It is also shown in [2] that set-valued quantiles characterize the distribution of a random set extension of the original random variable as well as its capacity functional.…”