# Control variate

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

## Example

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**.

## References

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