Suppose we have iid sample of size n from the distribution function of $F$ which has a continuous density $f$. How can I get the large sample joint distribution of p and q sample quantiles ?
Thanks for your help.
For every $1\leqslant k\lt\ell\leqslant n$, the joint density $g$ of the $(k/n,\ell/n)$ quantiles is $$ g(x,y)=cF(x)^{k-1}f(x)(F(y)-F(x))^{\ell-k-1}f(y)(1-F(y))^{n-\ell}\mathbf 1_{x\lt y}, $$ with $$ c=\frac{n!}{(k-1)!(\ell-k-1)!(n-\ell)!}. $$ To see this, note that the $(k/n,\ell/n)$ quantiles are roughly at $(x,y)$ when $k-1$ out of $n$ values of the sample are smaller than $x$, another one is roughly at $x$, $\ell-k-1$ out of the $n-k$ remaining ones are in $(x,y)$, another one is roughly at $y$ and the remaining $n-\ell$ ones are greater than $y$.