Control variates: Difference between revisions

Reworked →‎Underlying Principle: section
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(Reworked →‎Underlying Principle: section)
 
==Underlying Principle==
Let the [[Parameter#StatisticsStatistics_and_econometrics|parameter]] of interest be <math>\mu</math>, and assume we have a statistic <math>m</math> such that <math>\mathbb{E}\left[m\right]=\mu</math>. IfSuppose we are able to findcalculate another statistic <math>t</math> such that <math>\mathbb{E}\left[t\right]=\tau</math> andis <math>\rho_{mt}=\textrm{corr}\left[m,t\right]</math> area known values,value. thenThen
 
:<math>m^{\star}=m+c\left(t-\tau\right)</math>
 
is also [[bias of an estimator|an unbiased estimator]] for <math>\mu</math> for any choice of the constantcoefficient <math>c</math>. It can be shown that choosing
The [[variance]] of the resulting estimator <math>m^{\star}</math> is
 
:<math>\textrm{varVar}\left[(m^{\star}\right])=\textrm{Var}\left(1-m\rho_{mt}right) + c^2\,\textrm{Var}\left(t\right) + 2c\,\textrm{varCov}\left[(m,t\right])</math>;
:<math>c= - \frac{\sigma_m}{\sigma_t}\rho_{mt}</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}
:<math>\textrm{var}\left[m^{\star}\right]=\left(1-\rho_{mt}^2\right)\textrm{var}\left[m\right]</math>;
\textrm{Var}\left(m^{\star}\right) & =\textrm{Var}\left(m\right) - \frac{\left[{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>;
 
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>{Cov}\sigma_mleft(m,t\right)</math>, <math>\sigma_ttextrm{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==