Is this a valid approach to show an estimator is consistent?

Question

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?

Answer 1

Yes.

For the first part, unbiasedness means $\mathbb{E}[\hat{p}]=p$, together with $\mathit{Var}(\hat{p})=\mathbb{E}[(\hat{p}-p)^{2}]\to0$ implies $\hat{p}\overset{L_{2}}{\to} p$, which indeed is a stronger result than $\hat{p}\overset{\mathbb{P}}{\to}p$. Also, the law of large number directly gives consistency of $\hat{p}$.

The second part is simply invariance principle of MLE estimation.