 Finite-exposure Poisson inconsistency

Let independent counts satisfy $Y_i\sim \mathrm{Poi}(\theta w_i)$, with positive known exposures $w_i$, $\theta >0$, and $a_n={\sum }_{i=1}^n{w}_i\uparrow a_{\infty }<\infty$. Their likelihood is proportional to ${\theta }^{S_n}{e}^{-a_n\theta }$, where $S_n={\sum }_{i=1}^n{Y}_i$. If $S_n>0$, the positive maximum-likelihood estimator is $S_n/a_n$; when $S_n=0$, no maximum is attained on the open parameter space, while the closed-space maximizer is zero. This extended estimate converges almost surely to $S_{\infty }/a_{\infty }$, where $S_{\infty }\sim \mathrm{Poi}(\theta a_{\infty })$. Finiteness follows from $\mathbb{E}{S}_{\infty }=\theta a_{\infty }$, and the limiting Poisson distribution follows from the finite-sum laws. In particular, the probability of a zero estimate tends to $e^{-\theta a_{\infty }}>0$, proving failure of statistical consistency. The total Fisher information tends to $a_{\infty }/\theta$: infinitely many observations need not provide infinite information.