# Existence of estimator that reaches Cramer-Rao bound

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. does there always exist $\hat{\theta}$ such that $\text{Var}(\hat{\theta}) = \frac{1}{I(\theta)}$, $E(\hat\theta) = \theta$ (any standart regularity conditions are included)?

2. If there exist such an estimate, can it always be found by Maximum Likelihood Method (any regularity conditions are included)?

• The short answer is no. – Michael R. Chernick Dec 25 '16 at 22:18
• My answer was to a different question. As I recall prior to editing the question was just part 1. – Michael R. Chernick Dec 26 '16 at 19:27