Monte Carlo is a versatile computational method that may be used to approximate the means, μ, of random variables, Y , whose distributions are... Show moreMonte Carlo is a versatile computational method that may be used to approximate the means, μ, of random variables, Y , whose distributions are not known explicitly. This thesis investigates how to reliably construct fixed width confidence intervals for μ with some prescribed absolute error tolerance, "a, relative error tolerance, "r or some generalized error criterion. To facilitate this, it is assumed that the kurtosis, , of the random variable, Y , does not exceed a user specified bound max. The key idea is to confidently estimate the variance of Y 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 μ = E(Y ) can be used to develop a numerical integration method by writing the integral as μ = E(f(x)) = RRd f(x)⇢(x)dx, where x is a d dimensional random vector with probability density function ⇢. A similar idea is used to develop an algorithm for computing p = E(Y) where Y is a Bernoulli random variable. All of the algorithms have been implemented in the Guaranteed Automatic Integration Library (GAIL). Ph.D. in Applied Mathematics, May 2016 Show less
