Let $X_i \sim \mathrm{Poisson}(\lambda)$ for $i=1,2,\ldots,n$.
Let $\tau(\lambda) := (\lambda+1)e^{-\lambda}$ be the function we wish to estimate.
Find the UMVUE of $\tau$ and also check whether or not it attains the lower bound of the Cramer-Rao lower bound.
I found the UMVUE to be $\hat{\tau}(T) = (1-\frac{1}{n})^T (1-\frac{T}{n-1})$ where $T = X_1 + \ldots + X_n$.
(confirmed by showing it's unbiased).
I computed the Cramer Rao Lower bound to be $\lambda^3 \frac{e^{-2\lambda}}{n}$.
How would I compute the variance of $\hat{\tau}$? It seems like a really tedious calculation and I thought of Delta Method but that only involves the MLE of $\hat{\tau}$ (which asymptotically attains the C-R lower bound).
I computed the Variance of the UMVUE in WolframAlpha : http://www.wolframalpha.com/input/?i=-+(e%5E(-2k)(1%2Bk)%5E2)+%2B+sum+from+t%3D0+to+infinity+of+(1-1%2Fn)%5E(2t)+(1%2Bt%2F(n-1))%5E2+e%5E(-nk)(nk)%5Et%2Ft!
(where $k=\lambda$) and it seems to not attain it.