 Endpoint maximum likelihood for a singular location density

For density $r(x-\theta {)}^{r-1}$ on $[\theta ,\theta +1]$, $0<r<1$, the interior likelihood function increases without bound as $\theta$ approaches the sample minimum. The extended-likelihood estimator is that minimum. An arbitrary finite reassignment of the density at the endpoint can eliminate an attained finite maximum, so the endpoint convention must be stated.