Consider the zero-truncated Poisson distribution on the striclty positive integers, i.e. \begin{align} \mathbb{P}_{\theta}(X=k) = \frac{\theta^k}{k!(e^{\theta}-1)}\, \, \, , \, \, k=1, 2, ... \end{align} I show that the only unbiased estimator for $(1- e^{-\theta})$ is given by \begin{align} T(k) =0, \, \text{if $k$ is odd} \, \, \, \, \,\, \, \, \text{and} \, \, \, \, \, \, \, \, T(k) = 2, \, \, \text{if $k$ is even} \end{align}
I have to show that this estimator is absurd, but I don't see how to do it.
Any suggestions? Thanks in advance!