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

6  Antithetic variables I

\[ \]

Summary:

• Estimation is helped by combining individual estimates that are negatively correlated.

• For antithetic variables Monte Carlo estimation, we take pairs of non-independent variables \((X, X')\), to get the estimator \[ \widehat{\theta}_n^{\mathrm{AV}} = \frac{1}{n} \sum_{i=1}^{n/2} \big(\phi(X_i) + \phi(X'_i) \big) . \]

Read more: Voss, An Introduction to Statistical Computing, Subsection 3.3.2.

On Problem Sheet 1, you should now be able to answer all questions. You should work through this problem sheet in advance of the problems class on Thursday 15 October @ 11.