star-product of copulas

Question

I have recently come accross the star product of copulas, that is if $A$ and $B$ are 2-copulas and $\{C_t\}_{t\in[0,1]}$ is a family of copulas, then $C(x,y,z) = \int_0^y C_t(\frac{\partial}{\partial t} A(x,t),\frac{\partial}{\partial t} B(t,z))dt$ is the star product of $A$ and $B$, and $C$ itself is a copula.

Actually, I was wondering if we take two other 2-copulas $H$ and $G$, with $A\leq H$ and $B\leq G$ if then

$\int_0^y C_t(\frac{\partial}{\partial t} A(x,t),\frac{\partial}{\partial t} B(t,z))dt\leq \int_0^y C_t(\frac{\partial}{\partial t} H(x,t),\frac{\partial}{\partial t} G(t,z))dt\ \forall x,y,z\in[0,1]$

holds for all possible families $\{C_t\}_{t\in[0,1]}$. I am not too familiar with all the features of copulas so I am not sure if it follows directly from them.

Answer 1

Below is an example where the inequality does not hold:

For each $t$ we choose $C_t$ to be the independent copula $C_t(u,v)=u\cdot v$ and denote the corresponding star-product of $A$ and $B$ by $A\star B$.

For copulas $A$ and $B$ with probability density functions (pdf) $a$ and $b$, resp., we obtain from $$ A\star B(x,y,z)=\int_0^y \frac{\partial A}{\partial t}(x,t)\frac{\partial B}{\partial t}(t,z)\,dt $$ that the pdf of $A\star B$ is given by $$ a\star b(x,y,z) = a(x,y)\cdot b(y,z)\,. $$ Interpretation: Suppose that $A$ is a copula for random variables $X,Y$ and $B$ for $Y,Z$. If, conditioned on $Y$, the variables $X$ and $Z$ are independent, then $A\star B$ is a copula for $X,Y,Z$.

Example: Define $A,B,H,G$ by their pdfs $a,b,h,g$ on $[0,1]^2$ as follows:

• $a(x,y) = 1$ for all $x,y$,
• $b(y,z) = 0$ for $(y,z)\in [0,1/2]^2\cup(1/2,1]^2$ and $b(y,z) = 2$ otherwise,
• $h = 2 -b$,
• $g = b$.

Straightforward calculations yield that $A,B,H,G$ are indeed copulas and that the conditions $A\leq H$ and $B\leq G$ hold.

Observe now that $a\star b(x,y,z) = b(y,z)$ which implies that $A\star B(1/2,1,1/2)=1/4>0$. On the other hand, $h\star g$ is constant and equal to zero on the set $[0,1/2]\times[0,1]\times[0,1/2]$ and therefore $0=H\star G(1/2,1,1/2)<A\star B(1/2,1,1/2)$.