Edgeworth expansions for sampling without replacement from finite populations (Q1074258)

Let \(\{U_ 1,U_ 2,...,U_ N\}\) be a finite population, i.e., a finite collection of N distinguishable objects. Let \(u_ 1,u_ 2,...,u_ n\) denote a random sample of size n drawn without replacement from this population. Let X be a vector of k variables and \(X_ i\) the value of X for \(U_ i\). Let \(x_ i\) denote the X measurement on the sampled unit \(u_ i.\) Further, let \(F_ N\) be the distribution function on \(R^ k\) which assigns mass 1/N to each of the points \(X_ 1,X_ 2,...,X_ N\). Let \(p=n/N\) and \(q=1-p\). For a function f on \(R^ k\), \(E_ Nf\) is the mean of f under \(F_ N\); \(\Sigma_ N=E_ N(X-E_ NX)'\cdot (X-E_ NX)\). Let \[ Z_ n=(nq)^{-1/2}\sum^{n}_{i=1}(x_ i-E_ NX). \] It is known that a fairly large class of statistics used in sample surveys including several studentized functions of the multivariate sample means, can be expressed as \[ T_ n=\ell \cdot Z_ n+n^{-1/2}Z_ nLZ'\!_ n+o_ p(n^{-1/2}) \] where \(\ell \in R^ k\) and L is a \(k\times k\) matrix. In this paper, one-term Edgeworth expansions for the distributions of \(Z_ n\) and \(T_ n\), respectively, are obtained. An application of the results to the bootstrap is mentioned.