We introduce a new sufficient statistic for the population parameter vector by allowing for the sampling design to first be selected at random amongst a set of candidate sampling designs. In contrast to the traditional approach in survey sampling, we achieve this by defining the observed data to include a mention of the sampling design used for the data collection aspect of the study. We show that the reduced data consisting of the unit labels together with their corresponding responses of interest is a sufficient statistic under this setup. A Rao-Blackwellization inference procedure is outlined and it is shown how averaging over hypothetical observed data outcomes results in improved estimators; the improved strategy includes considering all possible sampling designs in the candidate set that could have given rise to the reduced data. Expressions for the variance of the Rao-Blackwell estimators are also derived. The results from two simulation studies are presented to demonstrate the practicality of our approach. A discussion on how our approach can be useful when the analyst has limited information on the data collection procedure is also provided.
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