= Solution
Set $\boxed{\beta=1/(1-\alpha)=1/r}$. For fixed $t>0$ and large $n$,
$$
\mathbb P\bigl(n^{1/r}(\widehat\theta_n-\theta)>t\bigr)
=\left(1-\frac{t^r}{n}\right)^n\longrightarrow\boxed{e^{-t^r}}.
$$
This <endpoint maximum likelihood for a singular location density> calculation gives a <Weibull distribution> with shape $r$. The estimator is a <consistent estimator>, and its error has order $n^{-1/r}$, faster than the $n^{-1/2}$ error of $\widetilde\theta_n$. Its <bias> is asymptotic to $\Gamma(1+1/r)n^{-1/r}$ and its <mean squared error> to $\Gamma(1+2/r)n^{-2/r}$, versus a positive constant times $n^{-1}$ for the sample-mean estimator. Thus \b[the minimum estimator is preferable for its much faster convergence], despite its positive finite-sample <bias>. This is a nonregular support-dependent model; ordinary regular-model efficiency bounds do not apply.
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