\[ \newcommand{\Exg}{\operatorname{\mathbb{E}}} \newcommand{\Ex}{\mathbb{E}} \newcommand{\Ind}{\mathbb{I}} \newcommand{\Var}{\operatorname{Var}} \newcommand{\Cov}{\operatorname{Cov}} \newcommand{\Corr}{\operatorname{Corr}} \newcommand{\ee}{\mathrm{e}} \]

5  Control variate

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Summary:

• Variance reduction techniques attempt to improve on Monte Carlo estimation making the variance smaller.

• If we know \(\eta = \Exg \psi(X)\), then the control variate Monte Carlo estimate is \[ \widehat{\theta}_n^{\mathrm{CV}} = \frac{1}{n} \sum_{i=1}^n \big(\phi(X_i) - \psi(X_i)\big) + \eta.\]

• The mean-square error of the control variate Monte Carlo estimate is \[{\displaystyle \operatorname{MSE}\big(\widehat{\theta}_n^{\mathrm{MC}}\big) = \frac{1}{n} \operatorname{Var}\big(\phi(X) - \psi(X)\big)}.\]

Read more: Voss, An Introduction to Statistical Computing, Subsection 3.3.3.