Suppose I have $n$ random variables $X_i \overset{iid}{\sim} f(x_i) $. Is it possible that the $r^{th}$ order statistic $X_{(r)}$ is independent of arbitrary random variable $X_i$? I really want this to be true, but my gut says no.
Follow-up. Is is possible to show $X_{(1)}-X_i$ is independent of $X_{(1)} -X_j$ where $i \neq j$. Now I believe this is true, but again, I am having a hard time being able to show why.