Given $X_1,X_2,...X_n\overset{\text{iid}}{\sim}F$, find the distribution of the sample inter quartile range, $F_n^{-1}(3/4)-F_n^{-1}(1/4)$ in terms of $F$ where, $F_n$ is the emperical distribution function defined as $F_n(x)=\frac{1}{n}\sum_i^n\mathbb{I}[X_i\le x]$ and $F^{-1}(t):=\inf\{x:F(x)\ge t\}$.
Theorem. For a random sample of size $n$ from an infinite population having value $x$ and density $f(x)$, the probability density at the $r$th order statistic $Y_{(r)}$ is given by $$g_r(y)=\frac{n!}{(r-1)!(n-r)!}\left[\int_{-\infty}^y f(x)dx\right]^{r-1}f(y)\left[\int_y^\infty f(x)dx\right]^{n-r}$$ In terms of $F$, I can rewrite $$g_r(y)=r{n\choose r}~[F(y)]^{(r-1)}~F'(y)~[1-F(y)]^{(n-r)}$$
If I use this to find the distribution of $Y_{(r)}$, I get $$G_r(y)=r{n\choose r}\int_{-\infty}^y[F(x)]^{(r-1)}~[1-F(x)]^{(n-r)}~dF(x)$$ For the $1/4$th quantile, I simply set $r=\lfloor n/4\rfloor$ where, $\lfloor\cdot\rfloor$ denotes the greatest integer function. Similarly, for the other end.
The final expression using this method isn't "nice-looking". I was wondering if there is an error in this method, or if there is an alternative method that results in a simpler form.
UPDATE. (From @JarleTufto 's comment)
Joint density of $r,s$th quantiles $$g_{r,s}(y_r,y_s)=\frac{n!}{(r-1)!(s-r-1)!(n-r)!}[F(y_r)]^{(r-1)}[F(y_s)-F(y_r)]^{s-r-1}[1-F(y_s)]^{(n-s)}F'(y_r)F'(y_s)$$ for $y_r\le y_s$. Transforming these to $Z_1=Y_r,Z_2=Y_s-Y_r$ we have joint density of $Z_1,Z_2$ $$f_{Z_1,Z_2}(z_1,z_2)=\frac{n!}{(r-1)!(s-r-1)!(n-r)!}[F(z_1)]^{(r-1)}[F(z_1+z_2)-F(z_1)]^{s-r-1}[1-F(z_1+z_2)]^{(n-s)}F'(z_1)F'(z_1+z_2)$$ for $z_2\ge0$. WE find the marginal distribution of $Z_2$ as $$f_{Z_2}(z)=\frac{n!}{(r-1)!(s-r-1)!(n-r)!}\int_{-\infty}^\infty [F(t)]^{(r-1)}[F(t+z)-F(t)]^{s-r-1}[1-F(t+z)]^{(n-s)}F'(t)F'(t+z)dt.$$ For IQR, we simply use $r=\lfloor n/4\rfloor,s=\lfloor 3n/4\rfloor$.
