 Likelihood supremum at an excluded boundary

A likelihood need not attain its supremum when the parameter space excludes a boundary point. For example, with parameter $0<\theta \le 1$ and a zero Poisson count, the likelihood $e^{-\theta }$ is strictly decreasing and has no maximum. Extending the parameter space to include zero gives an extended maximum-likelihood estimate there. A proposed estimator formula involving that boundary value must distinguish the extended estimate from an actual maximizer in the original space.