Rate of convergence for asymptotic variance of the Horvitz-Thompson estimator

Drawing distinct units without replacement and with unequal probabilities from a population is a problem often considered in the literature (e.g. Hanif and Brewer, 1980, Int. Statist. Rev. 48, 317-355). In such a case, the sample mean is a biased estimator of the population mean. For this reason, we use the unbiased Horvitz-Thompson estimator (1951). In this work, we focus our interest on the variance of this estimator. The variance is cumbersome to compute because it requires the calculation of a large number of second-order inclusion probabilities. It would be helpful to use an approximation that does not need heavy calculations. The Hajek (1964) variance approximation provides this advantage as it is free of second-order inclusion probabilities. Hajek (1964) proved that this approximation is valid under restrictive conditions that are usually not fulfilled in practice. In this paper, we give more general conditions and we show that this approximation remains acceptable for most practical problems. (C) 1998 Elsevier Science B.V. All rights reserved.