Let $X_1,...,X_n$ be the independent and identically distributed samples taken from an uniform distribution on $(a,b)$. $-\infty<a<b<\infty$. $X_{(1)},...,X_{(n)}$ are the order statistics of $X_1,...,X_n$.
How to prove that for all $a, b \in {\mathbf R}$, $$\frac{(X_{(i)}-X_{(1)})}{(X_{(n)}-X_{(1)})}$$ are independent $(i=2,...,n-1)$?
If $x>0$ is small enough then based on $X_{\left(1\right)}<X_{\left(2\right)}< X_{\left(3\right)}<X_{\left(n\right)}$:
$$P\left(\frac{X_{\left(2\right)}-X_{\left(1\right)}}{X_{\left(n\right)}-X_{\left(1\right)}}>x\wedge\frac{X_{\left(3\right)}-X_{\left(1\right)}}{X_{\left(n\right)}-X_{\left(1\right)}}<x\right)=0\neq P\left(\frac{X_{\left(2\right)}-X_{\left(1\right)}}{X_{\left(n\right)}-X_{\left(1\right)}}>x\right)P\left(\frac{X_{\left(3\right)}-X_{\left(1\right)}}{X_{\left(n\right)}-X_{\left(1\right)}}<x\right)$$
contradicting independence.