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)?


"Hey, Dave, how likely is it that my second-largest measurement is at least five?"


"What are you measuring?"


"Could be anything!"


"If you're measuring the number of meters between planets, then I'd say it's pretty likely. If you're measuring the number number of times one gives birth to triplets before age twenty, then I'd say it's pretty unlikely."


"...so how likely is it that my second-largest value is at least five?"

If you have no idea what the distribution is, then you shouldn't be able to know much about its statistics.
The same argument applies to simple statistics like the sample mean. The mean number of meters between planets is likely to be greater than five, while the mean number of times people give birth to triplets by age twenty is unlikely to be greater than five. The original distribution influences the distribution of the statistic.
