Problem Statement
We have two independent Poisson samples $X_1, \ldots, X_n$ with means $\lambda$ and $Y_1, \ldots, Y_n$ with means $\mu$. We would like to estimate $\tau=(\lambda-\mu)e^{-(\lambda+\mu)}$.
(a) Find a function of $X_1$ and $Y_1$ that is an ubiased estimator of $\tau$.
(b) Find the UMVUE of $\tau$.
(c) Calculate the asymptotic variance of this estimator when $\lambda=\mu$.
Context
I am studying some old exams, and I came across this problem. I think I have the correct unbiased estimator, but my UMVUE is definitely wrong since it is a function of unknown parameters. I am not sure if I made a mistake with my complete sufficient statistic, or I did the conditional expectation wrong.
Attempted Solution
(a) Find some function $g(X_1,Y_1)$ such that $\mathbb{E}(g)=\tau$.
$$
\tau = \lambda e^{-(\lambda+\mu)}-\mu e^{-(\lambda+\mu)}=\mathbb{P}(X_1=1, Y_1=0)-\mathbb{P}(X_1=0, Y_1=1)=\mathbb{E}\big[I_{X_1}(1)I_{Y_1}(0)-I_{X_1}(0)I_{Y_1}(1)\big],
$$
therefore, $U=I_{X_1}(1)I_{Y_1}(0)-I_{X_1}(0)I_{Y_1}(1)$ is unbiased for $\tau$.
(b) The Lehman-Scheffe theorem tells us that, given a complete sufficient statistic $T$ and an unbiased estimate $U$, the UMVUE is $E(U|T)$. Let $S_j=\sum_{i=j}^nX_i+Y_i$. From previous results we know that $S_j\sim Pois((n+1-j)(\lambda+\mu))$, where Poisson is in the exponential family of distributions and $S_1$ is a complete sufficient statistic. Therefore,
\begin{equation*}
\begin{split}
E(U|S_1=t)=& P(X_1=1,Y_1=0|S_1=t)-P(X_1=0, Y_1=1|S_1=t)\\
=& \frac{P(X_1=1,Y_1=0, S_2=t-1)-P(X_1=0,Y_1=1, S_2=t-1)}{P(S_1=t)}\\
\overset{ind.}{=}& \frac{P(X_1=1)P(Y_1=0)P(S_2=t-1)-P(X_1=0)P(Y_1=1)P(S_2=t-1)}{P(S_1=t)}\\
=&\bigg(\frac{n-1}{n}\bigg)^t\frac{t}{(n-1)(\lambda+\mu)}(\lambda-\mu),
\end{split}
\end{equation*}
is the UMVUE for $\tau$.