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

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?

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 $$\hat{p}$$.