Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 208 / 5 / a

Let $Y_i$ be iid with density $f$, set $H_i=h(Y_i)$, and put $I=E_{f}h(Y)$ and ${\tau }^2={\mathrm{Var}}_f(h(Y))$. The finite-second-moment assumption gives, for ${\tau }^2>0$,The sample variance $s_N^2=(N-1{)}^{-1}{\sum }_i(H_i-{\hat{I}}_N{)}^2$ consistently estimates ${\tau }^2$. The central limit theorem and Slutsky theorem therefore give the asymptotic confidence intervalIts confidence probability is approximately $1-\alpha$, conventionally written $100(1-\alpha )\%$; the printed $(1-\alpha )\%$ omits the factor $100$. For ${\tau }^2=0$, $h(Y)=I$ almost surely and the estimator has zero error. This interval concerns iid Monte Carlo estimators; correlated simulation output requires a long-run variance of a stationary process estimate instead of the iid sample variance.