Consider a probability distribution functions $f(x)$ with cumulative density function $F(x)$.
I would like to compute the PDF $g_{ij}$ of the gap $x_j-x_i$ where $x_i$ is the ith smallest sample.
The rth order statistics i.e the PDF of the rth smallest sample is $$\frac{n!}{(r-1)!(n-r)!}F(x)^{r-1}\left(1-F(x)\right)^{n-r}f(x)$$ where $n$ is the total number of samples. So I thought about convolving the ith order statistic of $f(-x)$ with the jth order statistic of $f(x)$ to obtain the PDF of the difference but this wont work because $x_i$ and $x_j$ are not independent variables. What would the strategy be ?