How can I find CSS for $\mu$ of $\text{Exp}(\mu,1),\;\mu\in\Bbb{R}$?
I just derived $X_{(1)}$ is a SS for $\mu\in\Bbb{R}$ where $X_{(i)}$ is the i-th order statistic
Now I'm struggling to show
$\int_{0}^{\infty}{g(y)e^{-ny}dy}=0\;\;\forall\mu\in\Bbb{R}\quad\Rightarrow\quad g\equiv0$
to prove
$E_{\mu}[g(X_{(1)})]=0\;\;\forall\mu\in\Bbb{R}$
I think it's true from the uniqueness of Laplace transform, but can I apply this property without any explanation on above statesment?
Since $n(X_{(1)}-\mu)\sim\text{Exp}(1)$ where $n$ is the sample size,
$E_{\mu}g(X_{(1)})=\int_{\mu}^{\infty}{ng(y)e^{-n(y-\mu)}dy}=0\;\;\forall\mu\in\Bbb{R}\quad\Rightarrow\quad g\equiv0$ from the Fundamental Theorem of Calculus (devide $e^{n\mu}$ and differentiate with $\mu$)