Underlying Principle of The CRN Technique
Suppose and are the observations from the first and second configurations on the jth independent replication.
We want to estimate
If we perform n replications of each configuration and let
then and Z(n) = Σ Zj / n is an unbiased estimator of .
And since the 's are independent identically distributed random variables,
In case of independent sampling, i.e., no common random numbers used then Cov(X1j, X2j) = 0. But if we succeed to induce an element of positive correlation between X1 and X2 such that Cov(X1j, X2j) > 0, it can be seen from the equation above that the variance is reduced.
It can also be observed that if the CRN induces a negative correlation, i.e., Cov(X1j, X2j) < 0, this technique can actually backfire, where the variance is increased and not decreased (as intended).
Read more about this topic: Variance Reduction
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