Control variates: Difference between revisions

m (Add control variates in the variance reduction category.)
The [[variance]] of the resulting estimator <math>m^{\star}</math> is
 
:<math>\textrm{Var}\left(m^{\star}\right)=\textrm{Var}\left(m\right) + c^2\,\textrm{Var}\left(t\right) + 2c\,\textrm{Cov}\left(m,t\right);.</math>
 
It can be shown that choosing the optimal coefficient
 
:<math>c^\star = - \frac{\textrm{Cov}\left(m,t\right)}{\textrm{Var}\left(t\right)}; </math>
 
minimizes the variance of <math>m^{\star}</math>, and that with this choice,
:<math>\begin{align}
\textrm{Var}\left(m^{\star}\right) & =\textrm{Var}\left(m\right) - \frac{\left[\textrm{Cov}\left(m,t\right)\right]^2}{\textrm{Var}\left(t\right)} \\
& = \left(1-\rho_{m,t}^2\right)\textrm{Var}\left(m\right);
\end{align} </math>
 
where
:<math>\rho_{m,t}=\textrm{Corr}\left(m,t\right); \, </math>
 
is the [[Pearson product-moment correlation coefficient|correlation coefficient]] of ''m'' and ''t''. The greater the value of <math>\vert\rho_{m,t}\vert</math>, the greater the [[variance reduction]] achieved.