A random sample $X_1, . . . , X_n$ of size n is taken from the Poisson distribution with parameter $\theta$. Let $X = (X_1, . . . , X_n)^T$ and $x = (x_1, . . . , x_n)^T$. It is proposed now to estimate the estimate the function of $\theta$, $g(\theta)$, where $g(\theta) = P(X = k) = exp(−\theta)\theta^k/k!$ for a specified integer k greater than zero.
An unbiased estimator of $g(\theta)$ is given by \begin{align*} U(X_j) = 1,\quad X_j = k \end{align*} \begin{align*} U(X_j) = 0,\quad \text{otherwise} \end{align*} for a fixed $j$ $(j = 1, . . . , n)$. On considering now without loss of generality the case $j = n$, \begin{align*} W(T) = E[U(X_n) | T] \end{align*} I know $W(T)$ is the UMVU estimator of $g(\theta)$, but I don't know how to prove that $W(T)$ is not a MVB (Minimum Variance Bound) estimator of $g(\theta)$. Could someone help me?