I'm trying to solve the following problem:
Let $X_1 , \dots , X_n \sim$ Bernoulli$(\theta)$, iid. Find $T_n$, the UMVUE of $\theta(1-\theta)$, and show that Var($T_n$) does not attain the Cramér-Rao lower bound.
My attempt to solve the problem was first find the sufficient statistic $\sum_{i=1}^{n}X_i \sim$ Binomial$(n,p)$.
Then, I found the UMVUE to be $T_n = \dfrac{n}{n-1}\bar{X}(1-\bar{X})$, where $\bar{X} = \dfrac{\sum_{i=1}^{n}X_i}{n}$.
The Cramér-Rao lower bound is: $\dfrac{(1-2\theta)^2}{n\left[\theta(1-\theta) \right]^{-1}}$
I got stuck on finding the Variance of $T_n$, since:
$Var(T_n) = \left(\dfrac{n}{n-1}\right)^2 Var(\bar{X}(1-\bar{X})) = \left(\dfrac{n}{n-1}\right)^2 \left[\dfrac{\theta(1-\theta)}{n} + \dfrac{Var((\sum_{i=1}^{n}X_i)^2)}{n^2}\right]$
Is there a simpler way to find $Var((\sum_{i=1}^{n}X_i)^2)$ than taking the 4th derivative of m.g.f. to get $E(\sum_{i=1}^{n}X_i)^4$?
Obs: this was a test question with limited time to solve.