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In the IID case, it is known that all order statistics are positively correlated.* Thus, we know that $$\text{Cov}(X_{(i)},X_{(j)}) \geq 0.$$ Is this known in the INID (independent, non-identically distributed) case? If it is not known, how could this be proved? If it does not seem true, what would be a counter-example?

  • E.g., see Bickel (1967), "Some contributions to the theory of order statistics"
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closed as off-topic by Bjørn Kjos-Hanssen, Andrey Rekalo, Stefan Kohl, Willie Wong, Chris Godsil Mar 12 '14 at 15:15

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You asked for an example with independent variables and negatively correlated order statistics. I can come close in two ways.

Here is an example with uncorrelated variables and negatively correlated order statistics.

enter image description here

Suppose the states of the world are the interval (-2,2), with variables $A = |x|-2$, $B = -1$, $C = +1$, $D = x+1-\text{sign}(x)$. Let $X=\{A,B,C,D\}$. Then $X_{(2)}$ and $X_{(3)}$ are negatively correlated.

Discretizing gives independent variables and uncorrelated order statistics.

Suppose there are four states of the world, and the four variables have values depending on the states as $A=(0,-2,-2,0), B=(-1,-1,-1,-1), C=(1,1,1,1), D=(0,2,0,2)$. Again let $X=\{A,B,C,D\}$. Then $X_{(2)}$ and $X_{(3)}$ are uncorrelated.

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If $X\sim N(0,1)$ and $(X_1,X_2) = (X,-X)$ the covariance of $(X^{(1)},X^{(2)})=\min(X_1,X_2),\max(X_1,X_2)$ is negative.

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    $\begingroup$ $X_1$ and $X_2$ are not independent in your example. $\endgroup$ – an12 Oct 25 '12 at 1:08
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    $\begingroup$ Try the FKG inequality, I think product measures are fine even if not i.i.d. and (I think) the order stats are increasing fctns of the data. $\endgroup$ – mike Mar 14 '13 at 17:28

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