Let $X\_{(1)}\leq X\_{(2)}\leq \cdots X\_{(n)}$ be the order statistics for a random sample from a continuous distribution with c.d.f. $F(x)$ and density $f(x)$. Define $U\_{i}$, $i=1,2\ldots,n,$ by 
$$U\_{i}=\frac{F(X\_{(i)})}{F(X\_{(i+1)})}, \quad i=1,\ldots,n-1, \: \mbox{and }\: U\_{n}=F(X\_{(n)}).$$


Find the joint distribution of $U\_{1},\ldots,U\_{n}$.


**Remark:** I need a suggestion or someone tell me in which book I can find this exercise or related theory that allows me to solve it.



Find the joint distribution of $U_{1},\ldots,U_{n}$ where $U_{i}=\frac{F(X_{(i)})}{F(X_{(i+1)})}$ and $X_{(i)}$ are order statistics
See the discussion after Theorem 1.2.9 in the book you refer to (Introduction to the Theory of Nonparametric Statistics, Randles & Wolfe).


If we let $V\_i=F(X\_{(i)})$ then it is explained that, because (by Theorem 1.2.9) $F(X)\sim U(0,1)$, we have that $V\_1\leq \cdots \leq V\_n$ are distributed as the order statistics from a uniform distribution on $(0,1)$. That is to say, they have joint density $g(v\_1,\ldots ,v\_n)=n!$ if $0<v\_1<\cdots <v\_n<1$, or $0$ otherwise.


To obtain the joint distribution of $U\_1,\ldots ,U\_n$ we just need to calculate the Jacobian of the transformation between the $U\_i$'s and the $V\_i$'s. The Jacobian is triangular in this case, so is quite straightforward to compute, giving the joint density for the $U\_i$'s as $f(u\_1,\ldots ,u\_n)=\prod\_{i=1}^n iu\_i^{i-1}$ if $0<u\_i<1$ for all $i$, or $0$ otherwise.


