Control variates: Difference between revisions

m
fixed formula formatting
(Reworked →‎Underlying Principle: section)
m (fixed formula formatting)
 
:<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>;
hence, the term [[variance reduction]]. The greater the value of <math>\vert\rho_{mt}\vert</math>, the greater the variance reduction achieved.
 
In the case that <math>\textrm{Cov}\left(m,t\right)</math>, <math>\textrm{Var}\left(t\right)</math>, and/or <math>\rho_{mt}</math> 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'''.
 
==Example==