Let $X_1,X_2,\ldots, X_n$ be $\operatorname{Normal}(\mu,\sigma^2)$

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 $\mu$.

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  • $\begingroup$ The independence of sample mean and sample variance under normal is also needed for this, correct? $\endgroup$ – user164144 Mar 9 '18 at 14:37
  • $\begingroup$ Nope. Since the sufficient statistics are complete, any function of them that is an unbiased estimate of a function of a parameter is the UMVUE of that function of a parameter. $\endgroup$ – aleshing Mar 9 '18 at 16:00

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