Upper-endpoint estimation in a truncated distribution ($\hat{M}=t_{(n)}$)

For a positive continuous density $f_{\theta }$ with strictly increasing cumulative distribution function $F_{\theta }$, observations from its upper-truncated law have likelihood ${\prod }_i{f}_{\theta }(t_i)/F_{\theta }(M{)}^n$ on $M>t_{(n)}$. Holding $\theta$ fixed, decreasing $M$ toward $t_{(n)}$ increases the likelihood; taking a supremum over $\theta$ preserves nonincreasing dependence on $M$. Provided the nuisance fit is finite, the usual closed-support endpoint estimate is $t_{(n)}$. With the literal open support $t<M$, only a limiting supremum need exist. A sample maximum is not a validated bound for future observations, and support depending on a parameter prevents automatic application of Wilks theorem.