Control variate

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.

Let the unknown parameter of interest be ${\displaystyle \mu }$, and assume we have a statistic ${\displaystyle m}$ such that the expected value of m is μ: ${\displaystyle \mathbb {E} \left[m\right]=\mu }$, i.e. m is an unbiased estimator for μ. Suppose we calculate another statistic ${\displaystyle t}$ such that ${\displaystyle \mathbb {E} \left[t\right]=\tau }$ is a known value. Then

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