 Solution

Set $\boxed{\beta =1/(1-\alpha )=1/r}$. For fixed $t>0$ and large $n$,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 ${\tilde{\theta }}_n$. 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 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.