Copula

A function that joins univariate distribution functions to form multivariate distribution functions. A 2-D copula is a function $C:I^2\to I$ such that

$\begin{displaymath} C(0,t)=C(t,0)=0 \end{displaymath}$

and

$\begin{displaymath} C(1,t)=C(t,1)=t \end{displaymath}$

for all $t\in I$ , and

$\begin{displaymath} C(u_2, v_2)-C(u_1, v_2)-C(u_2,v_1)+C(u_1, v_1)\geq 0 \end{displaymath}$

for all $u_1, u_2, v_1, v_2\in I$ such that $u_1\leq u_2$ and $v_1\leq v_2.$

See also Sklar's Theorem