Abstract
In this paper, we identify and characterize a family of sampling designs such that, under these designs, the sample median is a median-unbiased estimator of the population median. We first consider the simple random sampling case. A simple random sampling design has the median-unbiasedness property. Moreover, upon deleting samples from the simple random sampling case and imposing a uniform probability distribution on the remaining samples, the sample median is a median-unbiased estimator provided that the support meets a minimum threshold. However, there are other sampling designs, such as those based on balanced incomplete block designs, that do not need to meet the minimum threshold requirement to have the sample median be a median-unbiased estimator. We construct non-uniformly distributed sampling designs that have the median-unbiasedness property as well. In fact, the sample median is a best linear unbiased estimator within the class of linear median unbiased estimators. We show the sample median follows the Gauss-Markov Property under a simple random sampling design.
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Acknowledgements
We thank Hansheng Cheng for discussions and initial work leading to some aspects of this paper; we thank Cheng Ouyang and Raymond Mess for useful discussions. We thank the anonymous referees for their careful reading of the paper and for their numerous suggestions.
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A.S. Hedayat’s work was supported by U.S. National Science Foundation Grant #1809681. Jennifer Pajda-De La O has no conflict of interest.
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This work was partly supported by U.S. National Science Foundation Grant #1809681.
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Hedayat, A.S., Pajda-De La O, J. Median-Unbiasedness and the Gauss-Markov Property in Finite Population Survey Sampling. Sankhya A 83, 696–713 (2021). https://doi.org/10.1007/s13171-020-00221-4
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DOI: https://doi.org/10.1007/s13171-020-00221-4