# Is there no UMVUE for this case?

Let $X_1,X_2,... X_n$ be $Normal(\mu,\sigma^2)$

I seem to recall it said that there is no UMVUE for $\mu$ if $\sigma^2$ is also unknown but cannot find why this is so. Is this true?

The complete sufficient statistics for this model are $\bar{X}$ and $s^2$ (the sample mean and variance). Since we have $\mathbb{E}[\bar{X}]=\mu$, by Rao-Blackwell and its corollaries $\bar{X}$ is the UMVUE for $\mu$.