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.

[edit] 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 σm, σt, and/or ρmt 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.

[edit] References

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