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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)?

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

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