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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.

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    @Didier -- thank you for your very nice edits, and for pointing out the error (which I fixed per your suggestion). That is helpful to know even though this has been disproved.2011-08-14

1 Answers 1

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I have found a counterexample to your purported inequality $\frac{1}{2} ( \mathbb{E}(X_{2:2}) + \mathbb{E}(X) ) < \frac{1}{3} ( \mathbb{E}(X_{2:4}) + \mathbb{E}(X_{3:4}) + \mathbb{E}(X_{4:4}) ) = \frac{1}{3}( 4 \mathbb{E}(X) - \mathbb{E}(X_{1:4}) $.

Consider $X$ which follows $Beta(\alpha, 2)$. Then $\mathbb{E}(X) = \frac{\alpha}{2+\alpha}$ and $\mathbb{E}(X_{2:2}) = \frac{\alpha ( 4 \alpha+3)}{(2\alpha+1)(2\alpha+3)}$ and

$ \mathbb{E}(X_{1:4}) = \frac{\alpha ^4 \left(6912 \alpha ^5+34810 \alpha ^4+69601 \alpha ^3+68919 \alpha ^2+33734 \alpha +6524\right)}{(\alpha +2) (2 \alpha +1) (2 \alpha +3) (3 \alpha +1) (3 \alpha +2) (3 \alpha +4) (4 \alpha +1) (4 \alpha +3) (4 \alpha +5)} $

With this results, your inequalities is only satisfied when $\alpha > 0.51$.

In case you find this useful, here is Mathematica code performing this computations:

In[40]:= ineq =    With[{dist =       BetaDistribution[a,        2]}, (Expectation[x,          x \[Distributed] OrderDistribution[{dist, 2}, 2]] +         Mean[dist])/2 <      1/3 (4 Mean[dist] -         Expectation[x,          x \[Distributed] OrderDistribution[{dist, 4}, 1]])];  In[41]:= Reduce[ineq && a > 0, a, Reals] // N  Out[41]= a > 0.512761 
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    ok. it has benn undone.2011-08-14