Endpoint maximum likelihood for a singular location density
= 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.