Abstract
Ranked set sampling (RSS) for estimating a population mean μ is studied when sampling is without replacement from a completely general finite populationx=(x 1,x 2,...,x N )′. Explicit expressions are obtained for the variance of the RSS estimator\(\hat \mu _{RSS} \) and for its precision relative to that of simple random sampling without replacement. The critical term in these expressions involves a quantity γ=(x−γ)′Γ(x−μ) where Γ is anN × N matrix whose entries are functions of the population sizeN and the set-sizem, but where Γ does not depend on the population valuesx. A computer program is given to calculate Γ for arbitraryN andm. When the population follows a linear (resp., quadratic) trend, then γ is a polynomial inN of degree 2m+2 (resp., 2m+4). The coefficients of these polynomials are evaluated to yield explicit expressions for the variance and the relative precision of\(\hat \mu _{RSS} \) for these populations. Unlike the case of sampling from an infinite population, here the relative precision depends upon the number of replications of the set sizem.
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Prepared with partial support from the Statistical Analysis and Computing Branch. Environmental Statistics and Information Division, Office of Policy, Planning, and Evaluation, United States Environmental Protection Agency, Washington, DC under a Cooperative Agreement Number CR-821531. The contents have not been subjected to Agency review and therefore do not necessarily reflect the views of the Agency and no official endorsement should be inferred.
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Patil, G.P., Sinha, A.K. & Taillie, C. Finite population corrections for ranked set sampling. Ann Inst Stat Math 47, 621–636 (1995). https://doi.org/10.1007/BF01856537
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DOI: https://doi.org/10.1007/BF01856537