Median-Unbiasedness and the Gauss-Markov Property in Finite Population Survey Sampling
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Median-Unbiasedness and the Gauss-Markov Property in Finite Population Survey Sampling A. S. Hedayat and Jennifer Pajda-De La O University of Illinois at Chicago, Chicago, USA 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. AMS (2000) subject classification. Primary 62D05; Secondary 62G30. Keywords and phrases. Median-unbiasedness, Gauss-Markov property, Simple random sampling, Balanced incomplete block design
1 Introduction In Hedayat et al. (2019), it is shown that there is no unbiased estimator of the population quantiles. However, unbiasedness is not the unique criterion. Median-unbiasedness, first considered by Brown (1947), is a possible alternative. Brown (1947) introduced median-unbiasedness because it “accomplish[es] as much as the mean-unbiased requirement and has the additional property that it is invariant under one-to-one transformation.” For example, a meanunbiased estimator for the variance does not yield, on taking the square This work was partly supported by U.S. National Science Foundation Grant #1809681.
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A. S. Hedayat and J. Pajda-De La O
root, an unbiased estimator of the standard deviation. Median-unbiasedness, however, is preserved under such non-linear but monotonic transformations, i.e. φ (median(X)) = median (φ(X)). Although, as a trade-off, the operator of the median does not meet the linearity condition: median(aX + bY) = a median(X) + b median(Y). From a mathematical point of view, the only difference between regular unbiasedness and median-unbiasedness is operations. Switching the expectation operator E(·) to the median operator median(·), is the only change of the two definitions. Sometimes we also call regular unbiasedness as meanunbiasedness. Now we may ask a question: Is there a median-unbiased estimator of finite population quantiles? The answer is “yes”. From Hedayat et al. (2019) Example 1, we know that the sample median is a median-unbiased estimator of the population median under SRS(5, 3). In fact, we prove a general result for any simple random s
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