Estimation

For $N(\mu, \sigma^2)$, compare the MSEs of $S^2 = \frac1{n-1}\sum(X_i - \bar X)^2$ and $\hat\sigma^2 = \frac{n-1}nS^2$, given $\operatorname{Var}(S^2) = \frac{2\sigma^4}{n - 1}$.