Let $X_1, X_2,…,X_n$ be an i.i.d. random sample from $N(0, \sigma^{2})$.
a. Find the variance of $\hat{\sigma}^{2}_{MLE}$
So I found $\hat{\sigma}^{2}_{MLE}$ by taking the derivative of the log of the normal pdf function, but from there I am not sure how to proceed. $\hat{\sigma}^{2}_{MLE}$ comes out to $\frac{\sum_{i=1}^n X_i^{2}}{n}$. From there, would I do $\text{var}\left(\frac{\sum_{i=1}^n X_i^{2}}{n}\right)$ ? How do I compute this? Thanks!
Best Answer
Hint: If $Y_1,\ldots,Y_n$ are independent random variables and $a_1,\ldots,a_n$ are real constants, then $$ \mathrm{Var}\left(\sum_{i=1}^n a_iY_i\right)=\sum_{i=1}^n a_i^2\mathrm{Var}(Y_i). $$