would appreciate some help here please -
Question: Find the MSE for the MOM estimator of the variance $\hat{\sigma^{2}} = \frac{n-1}{n}S^{2}$ based on a random sample from a normal distribution.
My attempt:
MSE = Bias^2 + Variance.
I know that $S^{2}$ is unbiased for $\sigma^{2}$, so the bias part is simply $E(\frac{n-1}{n}S^{2} - S^{2}) = -\frac{1}{n}\sigma^{2}$
Var($\hat{\sigma^{2}}) = Var(\frac{n-1}{n}S^{2}) = (\frac{n-1}{n})^{2}Var(S^{2}) = (\frac{n-1}{n})^{2}(E(S^{4}) - E(S^{2})^{2})$
I know that $ E(S^{2})^{2}) = \frac{1}{n^{2}}\sigma^{4}$ from above. Can someone show me how can I obtain an expression for $E(S^{4})$ please?
Many thanks