I have two large sets A and B of (integer) numbers, both obtained with two different (and unknown) probability distributions $\rho_A$ and $\rho_B$. A (also large) set C contains a proportion $p$ of elements sampled from $\rho_A$ and a proportion $1-p$ sampled from $\rho_B$, hence $\rho_C = p\ \rho_A + (1-p)\ \rho_B$.
I have access to the three sets A, B and C and I want to estimate $p$.
The catch is, the distributions $\rho_A$ and $\rho_B$ seem to have a power-law tail. So normalization is impossible, I can't compute the mean... I don't know the exact form of the distributions either.
How can I extract the value of the mixture coefficient $p$ from my three populations $A$, $B$ and $C$ ?