Why does the MLE of the uniform distribution not satisfy a Central Limit Theorem?

Question

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!

Answer 1

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.