TITLE: Why does the MLE of the uniform distribution not satisfy a Central Limit Theorem?
QUESTION [0 upvotes]: For $X ~ U(0,\theta$)
The MLE of $\theta = \max{x_i}$.
Why does this not satisfy $\sqrt{n*I(\theta)} *( \max(x_i) - \theta) -> Z $
Where Z has a normal distribution?
I understand that $\max{x_i} < \theta$ which would make the bracketed term negative but I don't think this is the answer. Also, the sequence of estimators is consistent.
Thanks for any help!

REPLY [2 votes]: The distribution of extremal values is seldom normally distributed. The maximum of a bounded random variable will converge to a Type 3 Extreme Value Distribution  (call ed weibull-law in the link).
At the more basic level, the curvature of the log-likelihood at the extreme value is undefined, hence it will not converge to a quadratic log-likelihood.
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