Solution
= Solution
For $0\leq m\leq\theta_0$,
$$
\mathbb P_{\theta_0}(M_n\leq m)
=\left(\frac m{\theta_0}\right)^n.
$$
Hence, for $z\geq0$,
$$
\mathbb P\{n(\theta_0-M_n)>z\}
=\left(1-\frac z{n\theta_0}\right)^n
\longrightarrow e^{-z/\theta_0}.
$$
Thus
$$
\boxed{n(\theta_0-\widehat\theta)\xrightarrow d Z,
\qquad Z\sim\operatorname{Exp}(1/\theta_0).}
$$
This is the <endpoint asymptotics of the symmetric-uniform maximum likelihood estimator>.