= Uniform endpoint maximum-likelihood estimator
{title2=$\widehat\theta_n=\max_iY_i,\quad n(\theta-\widehat\theta_n)/\theta\xrightarrow{d}\operatorname{Exp}(1)$}
For independent $U[0,\theta]$ observations, the likelihood decreases as $\theta^{-n}$ on $[\max_iY_i,\infty)$, so its maximizer is the <sample maximum> almost surely. Its exact distribution is $\mathbb P_\theta(\widehat\theta_n\leq y)=(y/\theta)^n$ for $0\leq y\leq\theta$. 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.
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