Given: $\theta>0$, $X_1,...,X_n$ i.i.d. RV with
$p_{\theta}(x) = \frac{1}{\theta}, \text{where } 0 \leq x \leq \theta$
and
$p_{\theta}(x)=0, \text{else.}$
In the given exercise the MLE solution $L_X(\sigma)$ is
$L_X(\sigma)=\frac{1}{\sigma^n}1_{\{0\leq X_{(0)} \leq X_{(n)} \leq \sigma \}}$
with the order statistic, which I unterstand. But then they claim $\hat\theta = X_{(n)}$ if $\theta\geq X_{(n)}$ and zero else, without a lot of explanation. Can you explain me this solution?
I see that $\frac{1}{\sigma^n}$ is strictly decreasing for $\sigma>0$. But it seems to me that we cannot conclude without knowing if the maximum of the RV is $X_{(n)} < 1$ or $X_{(n)}\geq1$.
