Practical problems
There are two practical problems with the approximate variance formula for \(\hat {\theta} \),
\[ \Var(\hat {\theta})\; \approx \;- \dfrac 1 {n \times E\left[\frac {\large d^2\; \log\left(p(X \;|\; \theta)\right)} {\large d\;\theta^2} \right]} \]Avoiding the expected value
A numerical value for the approximate variance of \(\hat {\theta}\) can be found with a further approximation,
\[ \Var(\hat {\theta})\; \approx \;- \frac 1 {\ell''(\hat {\theta})} \](Justified in full version)
Its square root provides us with a numerical value for the standard error of the maximum likelihood estimator,
\[ \se(\hat {\theta}) \;\;\approx\;\; \sqrt {- \frac 1 {\ell''(\hat {\theta})}} \]This formula lets us find an approximate numerical value for the standard error of almost any maximum likelihood estimator — even when based on models in which the data are not a simple random sample.