I have the following task: Let $Y$ be a binomial random variable. Find the asymptotic distribution of the ML estimate and find the asymptotic distribution of the estimator $(p(1-p))^{\frac{1}{2}}$ of the standard deviation. What happens if $p=0.5$ ?
For $Y \sim Bin(n,p)$, the MLE of $p$ is $\hat{p}=\frac{Y}{n}$, that we obtain after differentiating the log-likelihood function and setting it to zero (likelihood equation). $$ \frac{\partial l(p)}{\partial p}=\frac{Y-np}{p(1-p)} \stackrel{!}{=} 0 $$ To compute the variance of the ML estimate, I compute the second derivative and take the expected value, so: $$ -\mathbb{E}\Bigl(\frac{\partial^2 l(p)}{\partial p} \Bigr) = \frac{n}{p(1-p)} $$ Taking the inverse gives the asymptotic variance of $\hat{p}$: $$ \mathbb{Var}(\hat{p})=\frac{p(1-p)}{n} $$
By the Central Limit Theorem, I know that $\sqrt{n}(\hat{p}-p) \xrightarrow{d} \mathcal{N}(0,p(1−p))$.
Do I now have to apply the delta method with $g(\hat{p})=\sqrt{\hat{p}(1-\hat{p})}$? The squared derivative of $g$ would be $\Bigl(\frac{1-2\hat{p}}{2\sqrt{\hat{p}(1-\hat{p})}} \Bigr)^2$. I thus get: $$ \sqrt{n}(\sqrt{\hat{p}(1-\hat{p})}-\sqrt{p(1-p)})\xrightarrow{d} \mathcal{N}\Bigl(0,p(1−p)\Bigl(\frac{1-2p}{2\sqrt{p(1-p)}} \Bigr)^2\Bigr) $$
where the last term is zero for $p=1/2$.
Is this so far correct? Also, what happens then for $p=0.5$?
