# Covariance of INID order statistics [closed]

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"

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. 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.

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
• 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 '13 at 17:28