# Cramer - Rao bound. Questions of existence and its likelihood properties.

There is a well known classical result called Cramer - Rao bound: https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Rao_bound Particularly, it is a lower bound for a variance of any unbiased estimate. The question is about to things:

1. Is there always an estimate which reaches this bound, i.e. is there always exist $\hat{\theta}$ such that, $var(\hat{\theta}) = \frac{1}{I(\theta)}$, $E(\hat\theta) = \theta$? (any standart regularity conditions are included)

2. If there exist such a estimate, is it always can be found by Maximal Likelihood Method?(as below any regularity conditions are included)

1. Generally, the answer is no. When you are talking about all estimators it is a little tricky, as this bound holds for asymptotically unbiased estimators. If you are restricted only to unbiased estimators (i.e., "estimable" parameters), then $1/p$, where $X\sim Ber(1,p)$, is an example of non-estimable function. If your class is broader - then MLE's invariance property can provide you with a good estimator of $1/p$.