Suppose we have $i$-th and $j$-th order statistics from a sample of i.i.d. random variables.

How to derive the difference between these order statistics? For example, it's obvious that, if $j>i$, then $F_j-F_i\geq0$. However, this does not apply to independent order statistics.

What would be the right way to find the distribution of $F_j-F_i$ provided that both values belong to the same sample?

P. S. I'm sorry if the question sounds awkward, my knowledge of probability theory is rather limited

  • $\begingroup$ For continuous variables, see stats.stackexchange.com/…. The answer is complicated for non-continuous variables due to the possibility of ties. There is no such thing, btw, as "independent order statistics:" by construction, the order statistics are not independent. $\endgroup$ – whuber Jan 24 '20 at 21:42
  • $\begingroup$ Yes, I must have made myself misunderstood in the description. What I meant is that a statistic of higher order from one sample is not necessarilty greater than another statistics of lower order from another sample -- even if the samples have the same size and data in both samples are identically distributed. However, that is not the case if both order statistics come from the same sample $\endgroup$ – Igor Yegin Jan 24 '20 at 22:00

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