 Solution

Write $S={\sum }_i{X}_i$. The likelihood function for the Poisson distribution is proportional to $e^{-n\theta }{\theta }^S$. For $S>0$, differentiation of the log-likelihood gives $-n+S/\theta =0$, and its second derivative is negative. For $S=0$, the likelihood decreases with $\theta$ and has its maximum at the boundary zero. HenceThe estimator is unbiased, since $\mathbb{E}\hat{\theta }=n^{-1}{\sum }_i\mathbb{E}{X}_i=\theta$. Its variance is $\theta /n$. If the parameter space excludes zero, an all-zero sample gives a supremum as $\theta \downarrow 0$ rather than an attained maximum.