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Let $X_1, · · · , X_n$ be a random sample from the uniform distribution on $[0, θ]$. I want to get the variance of the maximum likelihood estimator of $θ$ and check whether the variance decrease at the rate of $1/n$. Besides, whether the Cramèr-Rao information bound hold in this case.

For the first part, I got $$L(\theta) = \frac{1}{\theta^n} \mathbb I(X_{(n)} \leq \theta)$$

So, $$\hat \theta_{MLE} = \max(X_1,...,X_n)$$

For next step, I don't know how to calculate the variance of this MLE and don't know how to check the 2 questions. Any help or hint is welcome

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    $\begingroup$ The trick is to write down the density function of the maximum and either notice that its mean and variance are easy to compute or notice that it's a scaled Beta distribution. $\endgroup$
    – Thomas Lumley
    Mar 16, 2022 at 1:58
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    $\begingroup$ One thing often ignored when computing the CRB is the regularity conditions. The Fisher information must be well-defined and the expectation and differentiation should be interchangeable. The 2nd condition when the support is bounded requires the support to be independent of the parameter being estimated. $\endgroup$
    – Daeyoung
    Mar 16, 2022 at 3:26

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