An Improvement of probability approximation of randomized orthogonal array sampling

Abstract

Let X be a random vector uniformly distributed on [0, 1][superscript 3] and let ƒ be an integrable function from ℝ3 into ℝ and define µ = Eƒ(X) = ∫ƒ(x)dx. A simple estimator of µ is 1/n ∑_n(i=1)nƒ(Xi) where X₁, X₂, ..., Xn are independent random vectors and uniformly distributed on [0,1][superscript 3]. However, there are many methods to choose the points Xi's. One of those is the orthogonal array. In 1996, Loh was the first one who considered the normal approximation of W = µ-µ/√Var(µ) where Var (µ)>0 and gave a uniform bound. In 2008, Neammanee and Laipaporn improved the rate of convergence of Loh to be O(q-1/2) with the assumption that the sixth moment of ƒX is finite. In this thesis we improve their results under the finiteness of the fourth moment of ƒX. In the second part, we improve a non-uniform concentration inequality for a randomized orthogonal array which is given by Neammanee and Laipaporn in 2006.