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 $σ m$ , $σ t$ , 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.