Control variates: Difference between revisions

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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.<ref>Glasserman, P. (2004). ''Monte Carlo Methods in Financial Engineering''. New York: Springer. ISBN 0-387-00451-3 (p. 185)</ref>
==Underlying principle==
Let the unknown [[Parameter#Statistics_and_econometricsStatistics and econometrics|parameter]] of interest be <math>\mu</math>, and assume we have a [[statistic]] <math>m</math> such that the [[expected value]] of ''m'' is &mu;: <math>\mathbb{E}\left[m\right]=\mu</math>, i.e. ''m'' is an [[bias of an estimator|unbiased estimator]] for &mu;. Suppose we calculate another statistic <math>t</math> such that <math>\mathbb{E}\left[t\right]=\tau</math> is a known value. Then
:<math>m^\star = m + c\left(t-\tau\right) \, </math>
:<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,
{{refimprove|date=August 2011}}
* Ross, Sheldon M. (2002) ''Simulation'' 3rd edition ISBN 978-0-12-598053-1