Bounds in a combinatorial central limit theorem for randomized orthogonal array sampling designs

Abstract

Let X be a random vector uniformly distributed on [0, 1] [superscript 3] and let f be an integrable function from R [superscript 3] into R and define [mu] = Ef (X) = [the integral of][supscript [0,1][superscript 3]]f(x)dx. A simple estimator of [mu] is [mu]^ = 1/n sigma [superscript n][subscript i = 1]f(X[subscript i] where X[subscript 1], X[subscript 2],...,X[subscript n] are independent random vectors and uniformly distributed on [0, 1] [superscript 3]. However, there are many methods to choose the poins X[subscript i]'s. One of those is the orthogonal array. In 1996, Loh proved that [mu]^ obeys a central limit theorem and a uniform bound for the distribution of [mu]^ and normal distribution was given.In this thesis, we improve a uniform bound given by Loh and give a non-uniform bound using Stein's, method. Furthermore, we also establish a uniform and a non-uniform concentration inequality.