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.
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Underlying principle
Let the unknown parameter of interest be
is also an unbiased estimator for
It can be shown that choosing the optimal coefficient
minimizes the variance of
where
is the correlation coefficient of m and t. The greater the value of
In the case that
Example
We would like to estimate
using Monte Carlo integration. This integral is the expected value of
and U follows a uniform distribution [0, 1]. Using a sample of size n denote the points in the sample as
Now we introduce
Using
The variance was significantly reduced after using the control variates technique. (The exact result is