Guaranteed Conservative Fixed Width Confidence Intervals Via Monte Carlo Sampling

TL;DR

This paper presents a two-stage Monte Carlo algorithm for constructing guaranteed, conservative fixed-width confidence intervals for the mean of a random variable, assuming bounded kurtosis, with applications to multidimensional integration.

Contribution

It introduces a novel two-stage method that reliably constructs confidence intervals with prescribed error bounds under kurtosis assumptions, improving the robustness of Monte Carlo estimates.

Findings

The method guarantees confidence interval coverage under kurtosis constraints.

It effectively estimates variance from initial samples to determine sample size.

Applicable to multidimensional integrals with theoretical guarantees.

Abstract

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 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 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 multidimensional cubature.