Pareto Sampling versus Sampford and Conditional Poisson Sampling |
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Authors: | LENNART BONDESSON IMBI TRAAT ERS LUNDQVIST |
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Affiliation: | Department of Mathematics and Mathematical Statistics, UmeåUniversity; Institute of Mathematical Statistics, University of Tartu; Department of Mathematics and Mathematical Statistics, UmeåUniversity |
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Abstract: | Abstract. Pareto sampling was introduced by Rosén in the late 1990s. It is a simple method to get a fixed size π ps sample though with inclusion probabilities only approximately as desired. Sampford sampling, introduced by Sampford in 1967, gives the desired inclusion probabilities but it may take time to generate a sample. Using probability functions and Laplace approximations, we show that from a probabilistic point of view these two designs are very close to each other and asymptotically identical. A Sampford sample can rapidly be generated in all situations by letting a Pareto sample pass an acceptance–rejection filter. A new very efficient method to generate conditional Poisson ( CP ) samples appears as a byproduct. Further, it is shown how the inclusion probabilities of all orders for the Pareto design can be calculated from those of the CP design. A new explicit very accurate approximation of the second-order inclusion probabilities, valid for several designs, is presented and applied to get single sum type variance estimates of the Horvitz–Thompson estimator. |
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Keywords: | acceptance–rejection conditional Poisson sampling Horvitz–Thompson estimator inclusion probabilities Laplace approximation Pareto sampling πps sample Sampford sampling variance estimation |
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