Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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!

share|improve this question
1  
Salih, your question looks better now after the edit! :) –  Haskell Curry Mar 5 '13 at 9:36

1 Answer 1

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.

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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.