Conceptual question - Is it impossible to get a UMVUE for an estimator if another, unknown parameter is required to reach the Cramer-Rao Lower Bound?

Say I'm interested in estimating the $\sigma^2$ of a normal population, which has a Cramer-Rao Lower Bound of $\frac{2\sigma^4}{n}$ if my calculations are correct. I found that the uniformly minimum variance unbiased estimator of $\sigma^2$ should be $\frac{1}{n} \sum_{i = 1}^n (x_i - \mu)^2$, using the equality condition for the Cauchy-Schwarz Inequality. Does this mean that if the population mean is unknown, then I have no UMVUE?

I have verified that $\frac{1}{n-1} \sum_{i = 1}^n (x_i - \bar{X})^2$ does not reach the CRLB stated above (although it achieves the bound asymptotically).

• Sorry, I mean $\sigma^2$ is also supposed to be unknown... Commented Oct 20, 2016 at 6:32
• When $\mu$ and $\sigma$ are unknown, $\frac{1}{n-1} \sum_{i = 1}^n (x_i - \bar{X})^2$ is the UMVUE, even though it does not reach the Cramer-Rao lower bound. Which is a lower bound, not a minimal value that some estimator should reach. Commented Oct 20, 2016 at 6:36
• Addendum: Using the Cramer-Rao lower bound is a way to prove an estimator is UMVUE, not a necessary property of the UMVUE. Commented Oct 20, 2016 at 6:41
• I see, thanks. In that case, how can I show that $\frac{1}{n-1} \sum_{i = 1}^n (x_i - \bar{X})^2$ is the UMVUE? (Sorry but I don't seem to have the right to vote up comments in this SE.) Commented Oct 20, 2016 at 6:45
• You can use the Lehmann-Scheffé theorem. Commented Oct 20, 2016 at 10:14