Q:

Variance of variance of MLE

I am trying to find the variance of maximum likelihod estimate
$Var(s^2)$
where $ s^2 = \frac{1}{n} \sum^n_{i=1} (x_i-\bar{x})^2$
(model $X_i \sim N(\mu,\sigma^2)$).
I try getting
$Var(s^2) = Var(1/n \sum^n_{i=1} x_i) - Var(\bar{x}^2)$
$ = 1/n^2 \sum E(x^4_i)-E(x^2_i)^2- Var(\bar{x}^2)$
$= \frac{2\sigma^4+4\mu^2\sigma^2$}{n} -  Var(\bar{x}^2) $
and from here I can't getting further. How can I derive the variance of $\bar{x}$, and is it correct to to type $\mu$ or should I type $\bar{x}$ here?

A:

If $X_1,..., X_n $ are iis where each $X_i  \sim N(\mu, \sigma ^ 2)$, then 
$$
\sum_{i=1}^n \frac{(X_i - \bar{X}_n) ^ 2}{\sigma ^2 } \sim \chi^2_{(n-1)},
$$
hence 
$$
\operatorname{Var}(\frac{1}{n}\sum (X_i - \bar{X}_n) ^ 2 )=\frac{\sigma ^ 4}{n ^2} \operatorname{Var}( \sum_{i=1}^n \frac{(X_i - \bar{X}_n) ^ 2}{\sigma ^2 } ) = \sigma ^ 4 \frac{2(n-1)}{n^2}.
$$

