I have a question on computing MSEs. So X1...Xn are iid N(m, sigma^2). So sample variance is obviously s^2 = 1/(n-1)sum(xi-xbar), and mle is 1/sum(xi-xbar)^2. I know that s^2 is an unbiased estimator, while the mle is biased, but how do I compute the MSE's of each?

So I think I figured out that I should use the distribution of s^2 somehow: I know (n-1)s^2/(sigma^2) is distributed according to a chi-squared distribution of df=n-1. And I can get the MSE(s^2) = E(s^4) - 2E(s^2sigma^2) + E(sigma^4), but I'm not exactly sure how I can use the distribution to calculate E(s^4) and E(s^2sigma^2)......

I figured it out in the end.... so MSE(s^2) = 2sigma^4/(n-1), and MSE(sigma-hat^2)=2sigma^4(n-1)/n^2. It was much less complicated than I made it out to be. Thanks!