Uniform Bound on Normal Approximation of Latin Hypercube Sampling

N Rerkruthairat,Kritsana Neammanee

doi:10.5539/jmr.v3n2p98

Abstract

Loh (Loh, W.L, 1996b) established a Berry-Esseen type bound for $W$, the random variable based on a latin hypercube sampling, to the standard normal distribution. He used an inductive approach of Stein's method to give the rate of convergence $\frac{C_d}{\sqrt{n}}$ without the value of $C_d.$ In this article, we use a concentration inequality approach of Stein's method to obtain a constant $C_d.$

Highlights

A latin hypercube sampling (LHS) was introduced by McKay, Beckman and Conover in 1978 (McKay, M.D., 1979) as a tool to improve the efficiency of different important sampling method
After the original paper appeared, LHS has been widely used in many computer experiments
1. for all 1 ≤ i1, ..., id ≤ n, 1 ≤ j ≤ d, X j(i1, ..., id) = (i j − Ui1,...,id, j)/n, and X(i1, ..., id) = (X1(i1, ..., id), ..., Xd(i1, ..., id)); 2. ηk = (ηk(1), ηk(2), . . . , ηk(n)), 1 ≤ k ≤ d, are random permutations of {1, ..., n} each uniformly distributed over all the n! possible permutations; 3

Summary

Introduction

A latin hypercube sampling (LHS) was introduced by McKay, Beckman and Conover in 1978 (McKay, M.D., 1979) as a tool to improve the efficiency of different important sampling method. Ηk(n)), 1 ≤ k ≤ d, are random permutations of {1, ..., n} each uniformly distributed over all the n! Stein (Stein, M.L., 1987) showed that the asymptotic variance of μn is less than the asymptotic variance of an analogous estimator based on an independently and identically distributed sample. D}, let π j(ω) = η j+1(ω)(η1(ω)−1) and for each i1, ..., id ∈ {1, ..., n}, define

Results

Conclusion