Bounds on a normal approximation for latin hypercube sampling

Abstract

Let X be a random vector uniformly distributed on [0,1][superscript d] and let ƒ : [0,1][superscript d] → ℝ be an integrable function. An objective of many computer experiments is to estimate. μ = Eƒ(X)=∫[subscript 0,1][subscript d]ƒ(x)dx . Among numerical integration techniques, Monte Carlo methods are efficient and competitive for high-dimensional integration. The Monte Carlo's estimator for the integral μ is given by μ[subscript n] = 1/n Σ[superscript n][subscript i=1] ƒ (X[subscript i]) where X[subscript 1], X[subscript 2],...,X[subscript n] are random vectors on [0,1][superscript d] McKay, Beckman and Conover (1979) introduced Latin hypercube sampling(LHS) as an alternative method of generating X[subscript 1], X[subscript 2],...,X[subscript n]. In this work, we investigate normal approximation of error bounds in the distribution of μ[subscript n] based on a Latin hypercube sampling.