Guaranteed Conservative Fixed Width Confidence Intervals via Monte Carlo Sampling

Abstract

Monte Carlo methods are used to approximate the means, μ, 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 μ as n tends to infinity. This article explores how one can reliably construct a confidence interval for μ 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 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 μ. We discuss the important case where $Y = f(\boldsymbol{X})$ and $\boldsymbol{X}$ is a random d-vector with probability density function ρ. In this case μ can be interpreted as the integral $\int _{{\mathbb{R}}^{d}}f(\boldsymbol{x})\rho (\boldsymbol{x})\,\,\mathrm{d}\boldsymbol{x}$, and the Monte Carlo method becomes a method for multidimensional cubature.