I am working on a problem which asks me to discuss the efficiency of the MLE $\hat{\theta}$ given that $X_1,\ldots,X_n \sim_{iid} \operatorname{Beta}(\theta,1) $.

I was able to deduce that

$$\hat{\theta} = \frac{n}{-\sum_{i=1}^n \ln X_i}$$

and that the Rao-Cramer Lower Bound is

$$RCLB=\frac{\theta^2}{n}$$.

Since $E[\hat{\theta}]=\frac{n}{n-1}\theta$ the MLE is asymptotically unbiased, and I found that

$$Var[\hat{\theta}]= \frac{n^2\theta^2}{(n-1)^2(n-2)}$$.

What bothers me a little is that I was able to proove that the coefficient of $\theta^2$ is a value that is larger than $1 \over n$ when $1.57 < n$ so I see that this variance is indeed larger than the RCLB.

However, the fact that when $n=1$ and $n=2$ is undefined makes me a bit uneasy.

Is there something that needs to be considered in these cases or is there an assumption that I am missing?