Hi, I am preparing for my test. I have a question about sufficient statistics. More to follow later.

$\displaystyle X_1, \cdots, X_n$ \mbox{is a random sample with pdf:}$

$\displaystyle f(x|\mu, \sigma) = \dfrac{1}{\sigma} e^{-(x-\mu)/\sigma} \;,\;\mu<x<\infty\;,\;0<\sigma<\infty. \mbox{Find a two dimesional sufficient statistic for}(\mu, \sigma)$

The problem has been worked this way:

Let $\displaystyle x_{(1)}=min_i\;x_i. \mbox{Then the joint pdf is}$

$\displaystyle f(x_1,....., x_n)= \prod_{i=1}^n \dfrac{1}{\sigma} e^{-(x-\mu)/\sigma} \;I_{(\mu,\sigma)}(x_i)\;=\;\left(\dfrac{e^{\mu/\sigma}}{\sigma}\right)^n\cdot e^{-\sum_i x_i/\sigma}\;I_{(\mu,\sigma)}(x_1)$

and thus factorization theorem concludes that $\displaystyle (X_{(1)}, \sum_i X_i)$ is a suff stat for (mu,sigma).

I am conused in why we are taking the indicator function and why was the minmum order statistic used. Could anybody explain? Thank You.