Let $X$ be a continuous random variable with values ranging from 0 to 1.
Let $X_{kn}$ be the random variable representing the $k$th smallest order statistic of $n$ draws from $X$. Note that $X_{kn}$ is not a sample, but the marginal distribution of the statistic.
I begin with a simple specific case below. Then I will present the more complicated general case.
I need to prove:
$ \frac12\mathbb{E}(X_{22}+ X) < \frac13\mathbb{E}(X_{24}+X_{34}+X_{44}) $
In other words, that the average of $X$ and the max of 2 draws of $X$ is less than the average of the top 3 marginal order distributions of 4 draws of $X$.
Any help would be tremendously appreciated.
Also, the more general case is below:
Let $c$ denote some constant integer $\geq 2$.
Let $i$ denote some integer $ \geq 1$.
Prove that $\frac1c\mathbb{E}(X_{c^i,c^i}+ (c-1)X) < \frac1{c^{i+1}-c^i+1}\mathbb{E}\left(\sum_{k=c^i}^{c^{i+1}}X_{k,c^{i+1}}\right) $ And, preferably, that the magnitude of the inequality increases with $i$.
Also note: I posted a similar problem on stats.SE last week (still unanswered). This one is different enough to warrant its own question through.