I have $n$ independent Bernoulli trails $y_i\sim B(p)$. I found the MLE estimate to be $\hat{p}= \bar{y}$, the sample mean. I would like to show it is consistent too.
I'm using the result: If $T$ is unbiased estimator of $g(\theta)$ and $\text{Var}(T)\to 0$ as $n$ $\to \infty$, then $T$ is also consistent for estimating $g(\theta)$.
This is satisfied for $\bar{y}$, because $\text{Var}(\hat{p})= \frac{np(1-p)}{n^2}\to 0 $ as $n\to \infty$. So this means $\bar{y}$ is consistent.
Also, I want to find the MLE of its variance, ie, MLE of $p(1-p)$.
For this, I'm using this result: If $T$ is MLE estimate of $\theta$, then $g(T)$ is MLE estimate of $g(\theta)$.
This gives me, $\hat{p(1-p)} = \bar{y}(1-\bar{y})$ as the MLE estimate.
Is this accurate?