I am repeatedly drawing $n$ random samples where the first sample is drawn from $f(x, \theta_1)$ the second from $f(x, \theta_2)$, $\enspace \dots$ , and the $n^{\text{th}}$ from $f(x, \theta_n)$. Here $f$ is the logistic distribution with scale 1 and $\theta_1$ through $\theta_N$ are known location parameters. Next, I only continue with the $r^{\text{th}}$ largest of these samples. To me this seems like a rather unoptimized procedure as I'm drawing $n$ samples, sorting the $n$ samples, and discarding $n-1$ samples repeatedly. Instead I am wondering if I can sample directly from the distribution of the $r^{\text{th}}$ largest sample. I searched around and found some ideas closely related to the Bapat–Beg theorem.

The results closest to what I'm looking for is the following: Given $X_1, \enspace \dots, \enspace X_N$ independent and not necessarily identically distributed (i.n.n.i.d) samples the cdf of the $r^\text{th}$ sample is given by:

$ F_{r:n}(x) = \sum_{m=r}^n \frac{1}{m!(n-m)!} per[\mathop{\mathbf{F}(x)}\limits_m\qquad \mathop{\mathbf{1 - F}(x)}\limits_{n-m}] $

according to result 2.4$^1$. However, this is the cdf and to do inverse transform sampling I would need the inverse cdf or quantile function. Do (known) expressions of the inverse cdf/ quantile function of order distributions of i.n.n.i.d. rvs exist?


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