Suppose that $X$ is a discrete random variable with $P(X=1)=θ$ and $P(X=2)=1−θ$. Three independent observations of $X$ are made: $x_1=1,x_2=2,x_3=2$.
Find the MOM estimator of $θ$. What is the MOM estimate and its variance?
So I have found the MOM estimator to be $\hat{\theta}=2-\bar{X}$ and based on these 3 observations, the MOM estimate is $\hat{\theta}=2-\frac{5}{3}=\frac{1}{3}.$ I'm pretty confident with this part.
For the question asking to find the SE of this estimate, my first thought is:
$\sigma^2=\mathbf{E}[X^2]-\mathbf{E}[X]^2=...=\theta(1-\theta).$
Hence $\mathrm{Var}(\hat{\theta})=\mathrm{Var}(2-\bar{X})=\mathrm{Var}(\bar{X})=\frac{\sigma^2}{n}=\frac{\theta^2(1-\theta)^2}{n}$. So plug in the estimated value of $\hat{\theta}=\frac{1}{3}$ and $n=3$, I got $\mathrm{Var}(\hat{\theta})=\frac{2}{27}$.
However, after reading through this post, I came up with another answer as suggested by the poster:
Take $\mathrm{Var}(\bar{X})$ to be the sample variance, which is $\frac{1}{2}((1-\frac{5}{3})^2+2(2-\frac{5}{3})^2)=\frac{1}{3}$, thus $\mathrm{Var}(\hat{\theta})=\mathrm{Var}(\bar{X})=\frac{1}{3}$.
Could anyone help me figure out which one is wrong? The first one makes more sense to me, but I can't seem to find a loop-hole in the poster's explanation. Thanks in advance!