# Likelihood Cramér-Rao Bound.

How can I show the following necessary and sufficient condition?

An unbiased estimator $\hat{\theta}$ of $\theta$ achieves the Cramér-Rao Lower Bound if and only if $$\frac{\partial \log(L(\theta))}{\partial \theta} = I(\theta) \cdot (\theta - \hat{\theta}),$$ where $I(\theta)$ and $L(\theta)$ denote respectively the information and likelihood functions of a sample $(X_{1},X_{2},\ldots,X_{n})$ of i.i.d. random variables having a smooth pdf.

The ‘$\Longrightarrow$’ implication is clear, but I don’t know how to prove the ‘$\Longleftarrow$’ implication.

Thanks a lot!

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Salih, your question looks better now after the edit! :) –  Haskell Curry Mar 5 '13 at 9:36