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Monte Carlo methods are used to approximate the means, $\mu$, of randomvariables $Y$, whose distributions are not known explicitly. The key idea isthat 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 aconfidence interval for $\mu$ with a prescribed half-width (or error tolerance)$\varepsilon$. Our proposed two-stage algorithm assumes that the kurtosis of$Y$ does not exceed some user-specified bound. An initial independent andidentically distributed (IID) sample is used to confidently estimate thevariance of $Y$. A Berry-Esseen inequality then makes it possible to determinethe size of the IID sample required to construct the desired confidenceinterval for $\mu$. We discuss the important case where $Y=f(\vX)$ and $\vX$ isa random $d$-vector with probability density function $\rho$. 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 multidimensionalcubature.

Guaranteed Conservative Fixed Width Confidence Intervals Via Monte Carlo Sampling