Q. We are given $Y_1,Y_2,...Y_n$ as NID(0,$\sigma^2$). Does there exist any unbiased eastimator for $\sigma^2$ whose variance follows the CRLB for $\sigma^2$.
Considering $\tau(\sigma) = \sigma^2$. On calculation, I resolved the CR inequality equation to $var(\hat{\tau})\geq \dfrac{2\sigma^4}{n}$. Does there exist any unbiased estimator of $\sigma^2$ that has this value as its variance? Hinted in the question was to make use of the fact that $\Sigma_{i=1}^n Y_i^2/\sigma^2$ has a chi-squared distribution with n degrees of freedom and $\mu = n, var = 2n$. How can I use this fact to ascertain whether there does exist any estimator with variance = $\dfrac{2\sigma^4}{n}$ or not?
P.S. I am not well acquainted with chi-squared distribution.