Uniform endpoint maximum-likelihood estimator (${\hat{\theta }}_n={\max }_i{Y}_i,n(\theta -{\hat{\theta }}_n)/\theta \stackrel{d}{\to }\mathrm{Exp}(1)$)

For independent $U[0,\theta ]$ observations, the likelihood decreases as ${\theta }^{-n}$ on $[{\max }_i{Y}_i,\infty )$, so its maximizer is the sample maximum almost surely. Its exact distribution is ${\mathbb{P}}_{\theta }({\hat{\theta }}_n\le y)=(y/\theta {)}^n$ for $0\le y\le \theta$. This proves consistency and an endpoint error of order $1/n$, with the displayed exponential limit. Parameter-dependent support violates the usual regularity condition behind nondegenerate square-root-sample-size normal limits; the ordinary interior score is not centered.