On the non-recursive implementation of multistage sampling without replacement

Variance estimation in multistage sampling without replacement usually requires considerable computational effort. One option is to implement explicit formulas on a computer, at least for some specific sampling designs. This approach becomes quite cumbersome to handle beyond two stages, both from the formulation and computer implementation points of view. Another option is to provide a general method to compute variance estimates for any number of stages. Such an approach may involve data structures and estimators which are recursively defined. The solution we present in this article is intended to be both general and computationally efficient by relying on a full-iterative implementation. The definition of the estimators remains implicit as in the recursive approach, but is expressed in terms of recurrence relations translated into iterative algorithms. These algorithms rely only on (dense) array data structures. Moreover, most of the necessary computer memory is only used during preliminary steps and is not required when performing the statistical calculations. (C) 2021 The Author(s). Published by Elsevier B.V.