Proof Sample Variance is Minimum Variance Unbiased Estimator for Unknown Mean

Asked Apr 19 '20 at 22:41

Active Apr 19 '20 at 22:41

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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!

asked Apr 19 '20 at 22:41

Susy A.

$(\overline X,S^2)$ is complete sufficient for $(\mu,\sigma^2)$ and $E(S^2)=\sigma^2$ implies $S^2$ is UMVUE of $\sigma^2$. – StubbornAtom Apr 20 '20 at 06:27
But they are jointly sufficient, is that enough to prove that the sample variance by itself is MVU? – Susy A. Apr 20 '20 at 08:25
Thank you once more, now I get it! – Susy A. Apr 20 '20 at 09:05
https://stats.stackexchange.com/q/250917/119261 – StubbornAtom Nov 13 '21 at 15:57

Proof Sample Variance is Minimum Variance Unbiased Estimator for Unknown Mean

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