I have the following problem: Imagine a vector of data points in $R^D$ say $(x_1,...,x_n)$. For all those points we measure the distances to all the other points thus creating $n$ vectors each one of them consisting of $n-1$ distances. We refer to those as such $(r_1^1,...,r_{n-1}^1)$ the distances are ordered i.e $r_1^1$ is the distance from $x_1$ to its nearest neighbor. Similarly, we create all those vectors and the last one would be $(r_1^n,...,r_{n-1}^n)$ where $r_{n-1}^n$ is the distance of $x_n$ to its most distant neighbor. Now the problem is that i want to derive the distribution of an arbitrary ratio of two consecutive neighbours i.e the distribution of $\frac{r_k}{r_{k+1}}$ for any $k$ from $1$ to $n-1$. The problem obviously has to do with order statistic. Let's call the random variable that describes the first-order statistics of all those distances $Y_1$, the r.v for the second-order statistics $Y_2$ and so on until $Y_{n-1}$.
My main problem is to properly adapt the probability density function of order statistics as shown in the Wikipedia page in my problem, mainly i have troubles with the arguments and what $F_X(x)$ is in my case. After this step, I would try to derive the p.d.f of the ratio using the random variable transformation method. In essence, in the end, I want an expression for the $f_\frac{Y_k}{Y_{k+1}}$.
Any help is greatly appreciated.