Let $\{Y_{i}\}_{i=1}^{n}$ be a random sample of the random variable $Y_{i}\sim \mathcal{N}(0,\sigma^{2})$, we define the following estimators for $\sigma^{2}$
$U=\frac{1}{n-1}\sum_{i=1}^{n}(Y_{i}-\bar{Y})^{2} \quad V = \frac{1}{n(n-1)}\left( (n-2)\sum_{i=1}^{n}Y_{i}^{2} + (n\bar{Y})^{2}\right)$
Show that they have same variance.
First of all, I showed that they are both unbiased estimators of $\sigma^{2}$ and the complete statistic for this case is $\sum_{i=1}^{n}Y_{i}^{2}$ using the open set condition for the exponential family. However they aren't functions of the sample only through $\sum_{i=1}^{n}Y_{i}^{2}$. I don't think the person that wrote the exercise wanted the reader to calculate the variance directly, I'm guessing I missed something along the way. Thanks in advance for any help.