# Rao Cramèr Lower Bound problem

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

• 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. Commented Mar 16, 2022 at 1:58
• 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. Commented Mar 16, 2022 at 3:26