Uniform Bound on Normal Approximation of Latin Hypercube Sampling

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 convergencewithout the value of C d . In this article, we use a concentration inequality approach of Stein's method to obtain a constant C d .

Non–uniform bound of normal approximation for Combinatorial random sums

“…. By Theorem 2.2, there exists a constant C > 0 such that for all n ≥ 4 such that 1 + z ≤ n 1 14 , we have…”

“…and (1.1)- (1.3) hold. Then there exists a positive constant C such that for fixed z ∈ R with P (1 + |z| ≤ N 1 14 ) = 1, we have…”

“…Then there exists C > 0 such that for fixed z ∈ R and a positive integer n such that 1 + |z| ≤ n 1 14 and n ≥ 4, we have…”

“…for n ≥ n 0 := max{4, (1 + |z|) 14 }. To complete the proof, it suffices to show that Last, we will show that lim n→∞ EB r Nn (EB Nn ) r = 1 for all r > 0.…”

“…Assume that(3.1),(3.2),(3.5),(3.7) and(3.8) hold. Then for fixed z ∈ R and n ≥ 4 such that 1 + |z| ≤ n 114 , we have…”