How far are $d$-dimensional copulas with uniform $(d-1)$-marginals from (total) independence?

Abstract

We consider the family ${\mathcal{C}}_d^{\Pi_{d-1}}$ of all $d$-dimensional copulas whose $(d-1)$-dimensional marginals are all equal to the $(d-1)$-dimensional product copula $\Pi_{d-1}$ and tackle the natural question, `how far away' from the $d$-dimensional product copula $\Pi_d$ elements in ${\mathcal{C}}_d^{\Pi_{d-1}}$ can be. We provide definitive answers for both, the uniform metric $d_\infty$ as well as the stronger, conditioning-based metric $D_1$. The established results clearly indicate that the family ${\mathcal{C}}_d^{\Pi_{d-1}}$ is larger than one might expect.