# Standard error of Method of Moment estimator

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!

The first part is correct. Ok. The last one is wrong. The variance of the estimator is always unknown. One can estimate is using $$\hat{\theta}$$ but then it is random and varies for evey sample. Moreover, the variance of the estimator should converge to 0 as $$n\to \infty$$ (if not, the estimator would not be consitent and that is really not desirable)
There is just a mistype: You got: $$\sigma^2 = E[X^2] - E[X]^2 = \theta (1-\theta)$$ which is correct. But then $$SE_{\hat{\theta}} = \sqrt{Var(\hat{\theta})} = \sqrt{Var(2-\overline{X})} = \sqrt{\frac{\theta (1-\theta)}{n}}$$ where we use that $$X_1,\dots, X_n$$ are independent identically distributed.
So here I denoted by $$SE_{\hat{\theta}}$$ the standard error for the estimator $$\hat{\theta}$$ of $$\theta$$. So as you can see $$SE_{\hat{\theta}}$$ is unknown since $$\theta$$ is unknown, but one can estimate $$\theta$$ by $$\hat{\theta} = 2-\overline{X}$$ (stochastic). With our sample we have $$\hat{\theta} = 1/3$$ so an estimate of the standard error would be $$\hat{SE}_{\hat{\theta}} = \sqrt{\frac{\frac{1}{3} (1-\frac{1}{3})}{3}} = \sqrt{\frac{\frac{2}{9}}{3}} = \sqrt{2/27}.$$