# Covariance of INID order statistics

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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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.
$X_1$ and $X_2$ are not independent in your example. –  an12 Oct 25 '12 at 1:08