# Proof Sample Variance is Minimum Variance Unbiased Estimator for Unknown Mean

I am trying to prove that the unbiased sample variance is a minimum variance estimator. In this problem I have a Normal distribution with unknown mean (and the variance is the parameter to estimate so it is also unknown).

I am trying to solve it through the Cramer Rao bound, however I am not sure how to compute Fisher's information matrix or the covariance matrix for the estimator.

Thank you!

• $(\overline X,S^2)$ is complete sufficient for $(\mu,\sigma^2)$ and $E(S^2)=\sigma^2$ implies $S^2$ is UMVUE of $\sigma^2$. Apr 20, 2020 at 6:27
• But they are jointly sufficient, is that enough to prove that the sample variance by itself is MVU? Apr 20, 2020 at 8:25
• Thank you once more, now I get it! Apr 20, 2020 at 9:05
• stats.stackexchange.com/q/250917/119261 Nov 13, 2021 at 15:57