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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.2012-04-25
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    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.2012-04-25

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