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Control variate

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

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Example

Let the parameter of interest be $μ$ , 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 $μ$ 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 $\vert\rho_{tm}\vert$ , the greater the variance reduction achieved.

In the case that $σ m$ , $σ t$ , and/or $ρ m t$ 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

S. P. Meyn. Control Techniques for Complex Networks, Cambridge University Press, 2007. ISBN-13: 9780521884419. Online: http://decision.csl.uiuc.edu/~meyn/pages/CTCN/CTCN.html

Category: Monte Carlo methods