We investigate a Poisson sampling design in the presence of unknown selection probabilities when applied to a population of unknown size for multiple sampling occasions. The fixed-population model is adopted and extended upon for inference. The complete minimal sufficient statistic is derived for the sampling model parameters and fixed-population parameter vector. The Rao-Blackwell version of population quantity estimators is detailed. An application is applied to an emprical population. The extended inferential framework is found to have much potential and utility for empirical studies.
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