Control variate

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In Monte Carlo methods, one or more control variates may be employed to achieve variance reduction by exploiting the correlation between statistics.


Let the parameter of interest be , and assume we have a statistic such that . If we are able to find another statistic such that and are known values, then

is also unbiased for for any choice of the constant . It can be shown that choosing

minimizes the variance of , and that with this choice,


hence, the term variance reduction. The greater the value of , the greater the variance reduction achieved.

In the case that , , and/or are unknown, they can be estimated across the Monte Carlo replicates. This is equivalent to solving a certain least squares system; therefore this technique is also known as regression sampling.


  • Averill M. Law & W. David Kelton, Simulation Modeling and Analysis, 3rd edition, 2000, ISBN 0-07-116537-1