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Is the asymptotic distribution of the MLE of an unbiased estimator $N(\theta, \frac{1}{nI(\theta)})$?

So if you know the Cramer-Rao lower bound, you know the asymptotic distribution of the MLE of an unbiased estimator?

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What do you mean by "the MLE of an unbiased estimator"? The MLE of a parameter is an estimator of that parameter. An unbiased estimator of a parameter is also an estimator of the parameter. In some cases an MLE is unbiased; often it is not. One may speak of "the MLE of a parameter" or of "an unbiased estimator of a parameter", but "the MLE of an unbiased estimator" is another thing entirely---probably just a phrase with no particular meaning. –  Michael Hardy Apr 25 '12 at 20:45
    
The Cramér–Rao lower bound is a bound on the variance of an unbiased estimator (as opposed to an MLE or other possibly biased estimator). Some biased estimators have variances below the Cramér–Rao lower bound. However, you included the word "asymptotic". For that I think you may be right in well-behaved cases. However, I'm thinking the variance of the MLE for $\theta$ in the uniform distribution on $[0,\theta]$ may behave like $1/n^2$ rather than like $1/n$. I find some of this is not fresh in my memory. –  Michael Hardy Apr 25 '12 at 20:53

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