On the implementation of maximum entropy sampling with unequal probabilities and without replacement

Abstract

Sampling with maximum entropy offers robustness to statistical inference based on randomization theory. However, there were no comprehensive, practical guides explaining how to implement maximum entropy sampling for finite populations with unequal probabilities and without replacement. This article serves as both a toolkit and a reference guide for researchers and engineers, filling a gap in the literature. It links key formal results with ready-to-use algorithms that can be implemented in any procedural programming language. Maximum entropy sampling is straightforward when the sample size is allowed to vary. This is achieved via the Poisson sampling design, in which the sample size is a random variable distributed according to a Poisson binomial distribution. In contrast, the conditional Poisson sampling design, which is obtained by conditioning Poisson sampling on a fixed sample size, has long posed a significant challenge to statisticians.

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  A compendium of formal results for Poisson sampling, the Poisson binomial distribution, and conditional Poisson sampling is presented.
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  The computation of inclusion probabilities up to the second order is detailed for the conditional Poisson sampling, and the corresponding algorithms are provided.
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  Ready-to-use algorithms are provided for implementing Poisson sampling and the Poisson binomial distribution. For conditional Poisson sampling, the rejective, draw-by-draw, sequential, and exchange sampling algorithms are detailed.