Find the joint distribution of $U_{1},\ldots,U_{n}$ where $U_{i}=\frac{F(X_{(i)})}{F(X_{(i+1)})}$ and $X_{(i)}$ order static.

Question

Let $X_{(1)}\leq X_{(2)}\leq \cdots X_{(n)}$ be the order statics 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.

Answer 1

As noted in previous answer, $V_{(i)}=F(X_{(i)})$ are order statistics for independent Uniform samples $V_1,\ldots,V_n$, where $V_i=F(X_i)$. Joint density function for $(V_{(1)},\ldots, V_{(n)})$ is (see here) $$ f_{V_{(1)},\ldots, V_{(n)}}(v_1,\ldots,v_n)=\begin{cases}n!, & 0\leq v_1\leq\ldots\leq v_n\leq 1,\cr 0 & \text{otherwise}.\end{cases} $$ Let us find c.d.f. for $U_{1},\ldots,U_{n}$. For all $t_1,\ldots,t_n\in(0,\,1)$ $$ P(U_{1}<t_1,\ldots,U_{n}<t_n) = P\left(\frac{V_{(1)}}{V_{(2)}}<t_1,\ \ldots, \ \frac{V_{(n-1)}}{V_{(n)}}<t_{n-1},\ V_{(n)}<t_n\right) = $$ $$=P\left(V_{(1)} < t_1V_{(2)},\ \ldots, \ V_{(n-1)} < t_{n-1}V_{(n)},\ V_{(n)}<t_n\right) = $$ $$ =\int\limits_0^{t_n}dv_n\int\limits_0^{t_{n-1}v_n}dv_{n-1}\cdot\dots\cdot\int\limits_0^{t_3v_4}dv_3\int\limits_0^{t_2v_3}dv_2\int\limits_0^{t_1v_2}n!\,dv_1 = t_1t_2^2t_3^3\cdot\dots\cdot t_{n-1}^{n-1}t_n^n. $$

Answer 2

First, for any random variable $X$ with cdf $F$ which is invertible away from 0,1 (so density positive everywhere on the domain would do it), $F(X)$ is $\text{Uniform}(0,1)$. So let $V_i=F(X_i)$ and notice that $V_{(i)}=F(X_{(i)})$ as well. So you're looking at the ratio of ordered uniform random variables $V_{(i)}$. This reminds me of similar questions on Poisson-Dirichlet Distributions, but they're not ordering Uniform random variables and they have an infinite number of random variables. An indepth introduction into this can be found here and a simpler explanation of the distribution here, Section 3.

The best hope on finding more information like this is to maybe look at just the Dirichlet distribution, see if anyone has worked on the ratio of the ordering of its coordinates, or see if you can work through the Poisson-Dirichlet material and come up with your own proofs.