Let $X_1, X_2, \ldots, X_n$ be a sequence of i.i.d. r.v. with bounded range (say, the interval [0,1]), with cdf $F$. Let $Y_1 \geq Y_2, \ldots, \geq Y_n$ be the corresponding order statistics. My question is very simple:
Is the distribution of $Y_2+Y_3$ uniquely determining the parent distribution $F$ ? You can assume for simplicity that $F$ is absolutely continuous.
If the answer is yes: How do I prove it ?
If the answer is no: Can you pls provide a counterexample ?
This question is related to a problem in Auction Theory.
Thanks for your attention.