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

4  Monte Carlo error II: practice

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Summary:

• We can approximate confidence intervals for a Monte Carlo estimate by using a normal approximation.

• To get the root-mean-square error below \(\epsilon\) we need \(n = \Var(\phi(X))/\epsilon^2\) samples.

• We can use a two-step process, where a small “pilot” Monte Carlo estimation allows us to work out how many samples we will need for the big “real” estimation.

Read more: Voss, An Introduction to Statistical Computing, Subsections 3.2.2–3.2.4.