Guaranteed Conservative Fixed Width Confidence Intervals Via Monte Carlo Sampling

Monte Carlo methods are used to approximate the means, $\mu$, of random variables $Y$, whose distributions are not known explicitly. The key idea is that the average of a random sample, $Y_1,..., Y_n$, tends to $\mu$ as $n$ tends to infinity. This article explores how one can reliably construct a confidence interval for $\mu$ with a prescribed half-width (or error tolerance) $\varepsilon$. Our proposed two stage algorithm assumes that the \emph{kurtosis} of $Y$ does not exceed some user-specified bound. An initial independent and identically distributed (IID) sample is used to confidently estimate the variance of $Y$. A Berry-Esseen inequality then makes it possible to determine the size of the IID sample required to construct the desired confidence interval for $\mu$. We discuss the important case where $Y=f(\vX)$ and $\vX$ is a random $d$-vector with probability density $\rho$. In this case $\mu$ can be interpreted as the integral $\int_{\reals^d} f(\vx) \rho(\vx) \, \dif \vx$, and the Monte Carlo method becomes a method for multidimensional cubature.