Nonparametric Order Statistics - Does this Exist? I was reading about order statistics on Wikipedia [retrieved 29 June 2022]:

Apparently, if we have a sample with $k$ elements (e.g., $x_1, x_2, ..., x_k$) and assume a probability distribution for each of these $k$ elements - we can also determine the probability distribution for each of these individual $k$ elements.  It seems that the very definition of an order statistic is based on some underlying assumption of a probability distribution.
My Question: Does this mean that for some sample, it is fundamentally impossible to compute the distribution of the $k$th order statistics without some some choice of probability distributions (i.e., non-parametrically)?

This was formerly a comment, but per your comment, I expanded it a bit.
For a sample of size $n=2,$ there are two order statistics min and max. Obviously, the parent distribution influences the distributions of both. The distribution of the minimum of two standard normal random variables (which can take negative values) differs from the dist'n of the minimum of two standard exponential random (which is exponential with rate 2) and can never take negative values.  (Using, R for
brief simulation.)
set.seed(2021)
mn = replicate(10^6, min(rnorm(2))) # min of two std normals
summary(mn)
     Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
-5.331726 -1.107641 -0.544854 -0.563720  0.000805  3.415120 

me = replicate(10^6, min(rexp(2)))  # min of two exponentials
summary(me)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.0000  0.1442  0.3467  0.5000  0.6935  6.6056 

