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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!

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

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