Proving that $C(x_1,x_2)=\max\{x_1+x_2-1,0\}$ is a copula

Question

I have the following problem. I need to prove the copula property for the function $C(x_1,x_2)=\max\{x_1+x_2-1,0\}$, better known as the lower Fréchet-Hoeffding-Boundary.

A copula is defined as a function $C : [0, 1]^d \to [0, 1]$ if there exists a random vector $U = (U_1, . . . , U_d)$ with joint c.d.f. $C$ and uniform marginal distributions on $(0, 1)$.

A similar question was asked here. My problem is, I can't see how the two definitions are equal. How would I prove this with my definition? I don't really know where to start and would be glad for any help.

Answer 1

You've to start considering the definition of subcopula. A subcopula is a function $A$ such as:

• $Dom{A}=V_1 \times V_2$, with $V_1$ and $V_2$ subsets of $I=[0,1]$
• For every $x_1,y_1,x_2,y_2$ in I such that $x_1 \leq x_2$ and $y_1 \leq y_2$ we have $A(x_2,y_2)-A(x_2,y_1)-A(x_1,y_2)+A(x_1,y_1) \geq 0$.
• There is at least on element $a_1 \in V_1$ and an element $a_2 \in V_2$ such that $A(a_1,x_2)=A(x_1,a_2)=0$.
• $A(x_1,1)=x_1$
• $A(1,x_2)=x_2$

A copula $C$ is a subcopula $A$ defined in the domain $I \times I$. Now

$C(1,1)-C(1,y_1)-C(x_1,1)+C(x_1,y_1) \geq 0$. So we can write $C(x_1,y_1) \geq C(1,y_1)+C(x_1,1)-C(1,1) = y_1+x_1-1$. Because $C(x_1,y_1) \geq 0$ we can conclude that $C(x_1,y_1) \geq max\{y_1+x_1-1,0\}$.

Since a copula is a subcopula, this inequality holds for copulas. For every $(x_1,x_2) \in I \times I$ such that $x_2 \geq 1-x_1$ we have that $max\{y_1+x_1-1,0\}=x_1+x_2-1$. On the other hand, for every $(x_1,x_2) \in I \times I$ such that $x_2 \leq 1-x_1$, $max\{y_1+x_1-1,0\}=0$.

The lower bound is only valid in 2 dimensions. For extension have a look here Fréchet–Hoeffding lower limit copulas in higher dimensions.