The distributions $Y, X, Z$ and $W$ are related as follows:
$$Y_1 = X + Z$$
$$Y_2 = X + W,$$
that is $X$ (random variable) is a common factor to the random variables $Y_1$ and $Y_2$, which otherwise differ for the terms $Z$ and $W$ (random variables). I only know the realizations of $Y_1$ and $Y_2$.
I'm trying to use the method of moments to study the effects of $Z$ and $W$ on $Y_1$ and $Y_2$. I'm particularly interested in testing whether $Y_1$ and $Y_2$ belong to the same distribution, and on the marginal contribution of $Z$ and $W$.
Could you please help me? Thank you.
