Variance Reduction - Underlying Principle of The CRN Technique

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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