# Control variate

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

## Example

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 $\vert \rho _{tm}\vert$ , 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. 