Multivariate upper semilinear copulas

In the problem setting of constructing an n-copula given its diagonal section and all of its (n−1)-dimensional marginals, we introduce a new class of symmetric n-copulas, which generalizes the well-known class of bivariate upper semilinear copulas. These new upper semilinear n-copulas are constructed by linear interpolation on segments connecting the main diagonal of the unit hypercube [0, 1]n to one of its upper faces. We focus on the case where the (n−1)-dimensional marginals are upper semilinear (n−1)-copulas themselves, in which case the n-copula is actually constructed given its diagonal section and the diagonal sections of its k-marginals (k∈{2,3,…,n−1}). We provide necessary and sufficient conditions on these diagonal sections that guarantee that the upper semilinear construction method yields an n-copula. Several examples are provided.