Question: Suppose that $X$ is a discrete random variable with :
\begin{array}{|r|r|r|r|} \hline X & 0 & 1 & 2 & 3\\ \hline \mathbb P(X=x) &\frac{2}{3}\theta &\frac{1}{3}\theta &\frac{2}{3}(1-\theta) &\frac{1}{3}(1-\theta) \\ \hline \end{array} Where $0\leq\theta\leq1$ is a parameter. The following $10$ independent observations were taken from such a distribution: $\{3,0,2,1,3,2,1,0,2,1\}$
(a) Find the method of moments estimate of $\theta$.
(b) Find an approximate standard error for your estimate.
(c) What is the maximum likelihood estimate of $\theta$ ?
(d) What is an approximate standard error of the maximum likelihood estimate?
I figure out (a+b), $\theta$ using method of moment estimate and got, $\bar\theta=\frac{7}{3}-\frac{1}{2}\mathbb{E}(\bar{X})$ and $$Var(\bar\theta)=-\frac{1}{10}\theta^2+\frac{1}{10}\theta+\frac{1}{180}$$ Using the above information and with Log Likelihood function I also manage to find out (c), $MSE_{\bar{\theta}}=\frac{1}{2}$
for (d), I have no idea. What I only know that is, $$\sqrt{n}(\bar{\theta}-\theta_0)⟶N\left(0,\frac{1}{I(\theta_0)}\right)\text{ Asymptotic normality of MLE}$$ But I really don't understand it truly, and how to solve (d) (Using it or other way)?
(a)
Your MoM estimator doesn't look right. In fact, being
$$\mu=\frac{\theta}{3}+\frac{4}{3}(1-\theta)+ (1-\theta)=\frac{7}{3}-2\theta$$
You get
$$\hat{\theta}_{MM}=\Bigg[\frac{7}{3}-\overline{X}_{10}\Bigg]\frac{1}{2}$$
That is
$$\hat{\theta}_{\text{estimate}}=\frac{5}{12}$$
(b)
Variance of MoM estimator looks fine, anyway the question is not asking MSE but just an estimation of standard error , that is standard deviation of the statistic sample distribution: $\approx 0.173$
(c) and (d): do the same procedure with ML estimation. Given that you know the asymptotic distribution of MLE I think it is not difficult for you to conclude.