Given that $(\Bbb R_+^n,\mathcal B(\Bbb R_+^n),(\Bbb P_{\vartheta})_{\vartheta \geq0})$ and $X_1,\dots,X_n$ are iid random variables and for $i=1,...,n$ the density $$f_\vartheta(x)=e^{\vartheta-x}\mathbf{1}_{[\vartheta;\infty)}(x)=\begin{cases} e^{\vartheta-x} & \text{if $\vartheta \leq x$}\\ 0 & \text{otherwise} \end{cases} $$
I want to calculate the MLE for $\vartheta$.
My attempt:
$L(\vartheta|x_i)=\Pi f_\vartheta(x) \mathbf{1}_{[\vartheta;\infty)} = \Pi e^{\vartheta-x_i}\Pi\mathbf{1}_{[\vartheta;\infty)}=\Pi \frac{e^{n\vartheta}}{e^{x_i}}\Pi\mathbf{1}_{[\vartheta;\infty)}=e^{n\vartheta}\frac{1}{e^\sum x_i}\Pi\mathbf{1}_{[\vartheta;\infty)}=e^{n\vartheta}\frac{1}{e^{n\bar x_n}}\Pi\mathbf{1}_{[\vartheta;\infty)}=\Pi\mathbf{1}_{[\vartheta;\infty)}e^{n\vartheta}e^{-n\bar x_n}=\mathbf{1}_{\cap X_i}e^{n\vartheta-n\bar x_n}=\mathbf{1}_{min {Xi}}e^{n\vartheta-n\bar x_n}$
Is this wrong, correct, not enough ? Thx