= Upper-endpoint estimation in a truncated distribution
{title2=$\widehat M=t_{(n)}$}
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>.
Back to article page
