imagine you have a sample $X_1, X_2, \ldots, X_n$ from a random variable $X$, and another sample $Y_1, Y_2, \ldots, Y_m$ from a random variable $Y$. You know that $Y = \phi(X)$. For concreteness, say $Y = a_0 + a_1 X + a_2 X^2$. How can you estimate $a_0$, $a_1$ and $a_2$ from the samples?
You don't how $X$ or $Y$ are distributed, and your samples do not come in pairs. In fact, $n \neq m$.
I am stuck trying to solve this. Perhaps it is a well-known problem in the statistics community, but I am unable to find anything about it.