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, 1n 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 ź { 2 , 3 , ź , 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.