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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"

flag	asked Oct 24 at 18:59 orderstats 101●2

Gerard Letac · Answer 1 · 2012-10-25 01:03:34Z UTC

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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.

answered Oct 25 at 1:03

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

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. – mike Mar 14 at 17:28

Covariance of INID order statistics

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