| Monte Carlo is a versatile computational method that may be used to approximate the means of random variables whose distributions are not known explicitly. This thesis investigates how to reliably construct fixed width confidence intervals for means with some prescribed absolute error tolerance, relative error tolerance or some generalized error criterion. To facilitate this, it is assumed that the kurtosis of the random variable does not exceed a user specified bound. The key idea is to confidently estimate the variance of the random variable by applying Cantelli's Inequality. A Berry-Esseen Inequality makes it possible to determine the sample size required to construct such a confidence interval. When relative error is involved, this requires an iterative process. This idea for computing means can be used to develop a numerical integration method. A similar idea is used to develop an algorithm for computing means of Bernoulli random variables. All of the algorithms have been implemented in the Guaranteed Automatic Integration Library (GAIL). |
