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This paper treats some examples where likelihood-based inference for certain model parameters may produce empty confidence sets. The first example concerns epidemics, and the parameter of interest is the basic reproduction number R0, which is to be estimated from the final size of an epidemic in a finite population. The second example treats estimation of the mean of the offspring distribution in a branching process, based on observing the total progeny, i.e. the total number of individuals ever born in the branching process. The final example considers estimation of the linear drift in a Brownian motion, based on observing the first hitting time of some horizontal barrier.