# 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.

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.

• The problem is equivalent to finding the joint distribution of $(\frac{Y_{(1)}}{Y_{(2)}},\cdots,\frac{Y_{(n-1)}}{Y_{(n)}}, Y_{(n)})$ where $Y_1,\cdots,Y_n$ is a random sample from $U(0,1)$ population. As one knows the joint distribution of the $Y_{(i)}$'s, a simple change of variables $Z_i=\frac{Y_{(i)}}{Y_{(i+1)}}$ for $i=1,2,\cdots,n-1$ and $Z_n=Y_{(n)}$ does the job. Indeed, $Z_1,\cdots,Z_n$ are independent. Apr 13, 2018 at 18:35

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 $t_1,\ldots,t_n\in(0,\,1)$ $$P(U_{1}<t_1,\ldots,U_{n}<t_n) = P\left(\frac{V_{(1)}}{V_{(2)}}<t_1,\ \ldots, \ \frac{V_{(n-1)}}{V_{(n)}}<t_{n-1},\ V_{(n)}<t_n\right) =$$ $$=P\left(V_{(1)} < t_1V_{(2)},\ \ldots, \ V_{(n-1)} < t_{n-1}V_{(n)},\ V_{(n)}<t_n\right) =$$ $$=\int\limits_0^{t_n}dv_n\int\limits_0^{t_{n-1}v_n}dv_{n-1}\cdot\dots\cdot\int\limits_0^{t_3v_4}dv_3\int\limits_0^{t_2v_3}dv_2\int\limits_0^{t_1v_2}n!\,dv_1 = t_1t_2^2t_3^3\cdot\dots\cdot t_{n-1}^{n-1}t_n^n.$$
• I think the condition $0\leq v_{1}\leq \cdots \leq v_{n}$ over $f_{V_{(1)},\ldots,V_{(n)}}(v_{1},\ldots,v_{n})$ has influence in the process of integration, this implies that the distribution function must wrote by cases. Feb 23, 2017 at 3:41
• @DiegoFonseca, this inequalities are consequences of inequalities $0\leq v_1$, $v_1\leq t_1v_2$ and so on. Say, $v_1\leq t_1v_2$ implies $v_1\leq v_2$ since $t_1\leq 1$.
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.