What does knowing two pairwise copulas tell us about the third

Question

Say we have three random variables, which are all standard uniforms: $$ X \sim U(0,1), \\ Y \sim U(0,1), ~\text{and}~~~ Z ~ U(0,1) $$ If we know two of the pairwise copulas, $C_{XY}$ and $C_{YZ}$, what can be said about the third $C_{XZ}$ ?

I believe that $C_{XZ}$ isn't unique, but is it at least possible provide bounds on this copula?

Note that there is a trivial solution, which are the Frechet bounds:

$\overline{C_{XZ}}(x,z) = M(x,z) = min(x,z)$

$\underline{C_{XZ}}(x,z) = W(x,z) = max(x + z - 1, 0)$

Is it possible to get tighter bounds? Does the solution change at all if we add a forth variable?

Answer 1

To answer your title question: Nothing in general. See also Durante et al. 2014.

REFERENCES:

• Durante, Fabrizio, Roger B. Nelsen, etc. "Pairwise and global dependence in trivariate copula models." In International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, pp. 243-251. Springer, Cham, 2014.