Normal copula vs Fréchet copula I'm trying to understand how to generate $2$ random variables $U_1,U_2$ which are correlated using the above $2$ methods with a rank correlation $P_s = 0.3$.


*

*First I create $2$ random variables $X_1,X_2$ from $U(0,1)$

*$Y_1 = X_1$ and $Y_2 = X_1(P_s) + X_2 \sqrt{(1-Ps^2)}$

*Now my $U_1$ and $U_2$ are Inverse $(Y_1,Y_2)$. 


(by Inverse it is meant inverse of normal distribution)
The above believe is from the normal copula. 
The Fréchet copula is given by
$$C(U_1,U_2) = p \,\min(U_1,U_2) + (1-p)\,\, \max (0,U_1+U_2-1)$$ With limited background in mathematics. I'm not quite sure how to do this, I assume I just use my $P_s$ value in place of $p$ but I'm not sure what the min and max term actually means. 
Thanks for any help.

the min (max) function just returns the minimum (maximum) of the two arguments.
$C(U_1,U_2)$ for the Fréchet copula is simply a linear ramp for zero probability at $(U_1=0,U_2=0)$ up to $(U_1=1, U_2=1)$ along the line $U_1=U_2$ (zero elsewhere) when $p=1$. for the opposite extreme, when $p=0$, $C$ is zero for $U_2<(1-U_1)$ and ramps linearly toward the $(U_1=1,U_2=1)$ corner.
if you sample many pairs of $U_1$ and $U_2$ using the first method and compute empirical CDF of $U_1$ and $U_2$ you can then compare that to Fréchet formula 
in order to synthesize $U_1$ and $U_2$ directly from the Copula, you would need to take the inverse solving for $(U_1,U_2)=C^{-1}(U[0,1])$, but for a given draw from $U[0,1]$ you'll have a continuous line of iso-C of $(U_1,U_2)$ pairs to select from
