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Given that $X$ is random matrix ($X'$ is transposed matrix) and $\Omega$ is invertible square matrix is inequality $$EX'X\cdot(EX'\Omega X)^{-1}\cdot EX'X\leq EX'\Omega^{-1}X$$ correct? All the expectations as well as inverse of $EX'\Omega X$ exist.

Maybe (but it is probably wrong)

$$EX'X\cdot(EX'\Omega X)^{-1}\cdot EX'X \leq EX'X\cdot(X'\Omega X)^{-1}\cdot X'X $$

Then I somehow need to show that

$$X'X\cdot(X'\Omega X)^{-1}\cdot X'X = X'\Omega^{-1}X$$

Maybe expressing $\Omega$ as $\Omega^{1/2}\cdot \Omega^{-1/2}$ might somehow work.

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do you mean for X to be a square matrix also ? If not, try the case where X is $n \times 1$ vector of i.i.d.s –  mike Nov 7 '12 at 14:36
    
unfortunately $X$ is not necessarily square –  jem77bfp Nov 7 '12 at 15:07
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