Let ${ \left\{ X_{ i } \right\} }_{ i=1 }^{ n } \sim E(a,\theta)$ where $a \in {\rm I\!R}$, and $\theta > 0$. Show that the smallest order statistic, $X_{(1)}$, has the exponential distribution $E(a,\theta/n)$ and that $2\sum _{ i=1 }^{ n }{ (X_{ i }-X_{ (1) }) } /\theta \sim { \chi }_{ 2n-2 }^{ 2 }$.
I have shown that the smallest order statistic $X_{(1)} \sim E(a,\theta/n)$. But I'm struggling in the second part. I think I can use the Basu Theorem, but I believe that there are other simpler option. Any hint?
