# 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

If θ^ reaches the Frechet-Cramer-Rao lower boundary it means it's efficient of a parametric function h(θ). If it is efficient, Var[T] reaches the mentioned bound, and therefore d/dθ ln(L(θ))= l(θ)*(T-E[T]).

It is a double way implication, really. Just remember that l(θ)=I(θ) when θ^ is unbiased of θ.

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i am asking the other implication.. efficient is the definiton... – Salih Ucan Mar 5 '13 at 18:32