The control variates method is a variance reduction technique used in Monte Carlo methods. It exploits information about the errors in estimates of known quantities to reduce the error of an estimate of an unknown quantity.

Underlying principle

Let the unknown parameter, and assume we have a statistic such that the expected value of m is , i.e. m is an unbiased estimator for is a known value. Then

. The variance is

, it can be shown that choosing the optimal coefficient

. (Note that this coefficient is the same as the coefficient obtained from a linear regression.) With this choice,

where

is the correlation coefficient, the greater the variance reduction achieved.

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.

is close to unity.

Example

We would like to estimate

using Monte Carlo integration, where

and U follows a uniform distribution[0,1]. Using a sample of size n. Then the estimate is given by

and combine the two into a new estimate

we obtain the following results.)

See also

• Antithetic variates

• Importance sampling

Notes

References

• Ross, Sheldon M. (2002) Simulation 3rd edition

• Averill M. Law & W. David Kelton (2000), Simulation Modeling and Analysis, 3rd edition.

• S. P. Meyn (2007) Control Techniques for Complex Networks, Cambridge University Press. . Downloadable draft (Section 11.4: Control variates and shadow functions)