Consider $U_i \sim^{iid} Bernoulli(\pi)$. Also consider:
$$Y_i | U_i = 0 \sim exp(1/\gamma) \text{ and } Y_i | U_i = 1 \sim exp(1/2\gamma) $$
What are the method of moment estimators of $\pi \text{ and} \text{ }\gamma$ ?
Here is my solution:
$E(U_i) = \pi \Rightarrow \hat{\pi_{MOM}} = \bar{U} = \sum U_i / n$
$E(Y_i) = E(E(Y_i | U_i)) = \gamma (1 + \pi) \Rightarrow \hat{\gamma_{MOM}} = \frac{\sum Y_i / n}{1 + \hat{\pi_{MOM}}}$
Are the estimates above right?
