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Set $S_n={\sum }_{i=1}^n{Y}_i$ and $a_n={\sum }_{i=1}^n{2}^{-i}=1-2^{-n}$. For the Poisson distribution observations, factors independent of $\theta$ can be removed from the likelihood function, leavingWhen $S_n>0$, the log-likelihood derivative is $S_n/\theta -a_n$ and its second derivative is negative. The unique positive maximum-likelihood estimator is thereforeThere is a genuine boundary qualification: if $S_n=0$, the likelihood decreases strictly on $\theta >0$, so no maximum is attained in the stated open parameter space. On the closure $\theta \ge 0$, its maximizer is $0$. Defining the usual extended maximum-likelihood estimator by the same displayed formula for every $S_n$ allows its statistical consistency to be examined; the all-zero event cannot be ignored because its probability does not tend to zero.

Indeed, sums of independent Poisson distribution variables give $S_n\sim \mathrm{Poi}(\theta a_n)$, soThus the extended maximum-likelihood estimator is inconsistent. More precisely, $S_n$ increases to $S_{\infty }={\sum }_{i\ge 1}{Y}_i$, and $\mathbb{E}{S}_{\infty }=\theta <\infty$ shows that this total count is finite almost surely. Its distribution is Poisson distribution with mean $\theta$, by taking limits of the finite-sum laws. HenceThis finite-exposure Poisson inconsistency leaves a nondegenerate random limit even with infinitely many observations. Correspondingly, the total Fisher information is $a_n/\theta \to 1/\theta$, rather than diverging. The common-distribution hypothesis in the introductory maximum likelihood estimation theorem is absent here.