Control variates

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 parameter of interest be $\mu$, and assume we have a statistic $m$ such that $\mathbb {E} \left[m\right]=\mu$. If we are able to find another statistic $t$ such that $\mathbb {E} \left[t\right]=\tau$ and $\rho _{mt}={\textrm {corr}}\left[m,t\right]$ are known values, then

$m^{\star }=m+c\left(t-\tau \right)$

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

$c=-{\frac {\sigma _{m}}{\sigma _{t}}}\rho _{mt}$

minimizes the variance of $m^{\star }$, and that with this choice,

${\textrm {var}}\left[m^{\star }\right]=\left(1-\rho _{mt}^{2}\right){\textrm {var}}\left[m\right]$;

hence, the term variance reduction. The greater the value of $\vert \rho _{mt}\vert$, the greater the variance reduction achieved.

In the case that $\sigma _{m}$, $\sigma _{t}$, and/or $\rho _{mt}$ 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.

See also

• Variance reduction
• Antithetic variates

References

• Ross, Sheldon M. Simulation 3rd edition ISBN 978-0125980531

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

• S. P. Meyn. Control Techniques for Complex Networks, Cambridge University Press, 2007. ISBN 9780521884419. Online: https://netfiles.uiuc.edu/meyn/www/spm_files/CTCN/CTCN.html