The maximum likelihood estimate of a Bernoulli process is simply given by $\hat{\theta}=\frac{\sum X_i}{N}$, where N is the total number of bernoulli trial and $X_i$ is the outcome of each trial.
This is an unbiased estimator and the variance of this estimator can be easily computed to be $Var(\hat{\theta}) = \frac{\theta(1-\theta)}{N}$. However, the actual $\theta$ is unknown.
So how do we estimate the variance of the estimator then ? Also, I would like an unbiased estimate of this variance. Its possible that this question has already been asked. If someone can give a pointer that would be great.
Thanks.