This is an open question: given a sequence of $n$ real numbers $x_1<x_2<\dots<x_n$, does there always exist a probability distribution, such that $\{x_i\}$ happens to be the $n$ *expected order statistics* of this distribution?

In other words, can we always "reverse engineer" the distribution from its expected order statistics? Note that there is **no restrictions** on the distribution, i.e., it can be continuous, discrete, or whatever. I wonder if anything is known regarding this existence problem.

(Edited 09/20/2017)

When sequence $\{x_i\}$ is unrestricted, the answer to above claim is **no**. This is shown by @Mateusz Kwaśnicki when $n=4$. However, under $n=4$, suppose $\{x_i\}$ satisfies the condition that $3(x_4−x_1)⩽7(x_3−x_2)$, then is there a method that can construct the distribution for which $\{x_1,x_2,x_3,x_4\}$ are the expected order statistics?

In other words, suppose $\{x_i\}$ satisfies the necessary conditions to be expected order statistics, is there a method to "reconstruct" the underlying distribution? Or is this too much to ask for?

4more comments