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Environmental and Ecological Statistics

, Volume 19, Issue 3, pp 309–326 | Cite as

Nonparametric mean estimation using partially ordered sets

  • Jesse FreyEmail author
Article

Abstract

In ranked-set sampling (RSS), the ranker must give a complete ranking of the units in each set. In this paper, we consider a modification of RSS that allows the ranker to declare ties. Our sampling method is simply to break the ties at random so that we obtain a standard ranked-set sample, but also to record the tie structure for use in estimation. We propose several different nonparametric mean estimators that incorporate the tie information, and we show that the best of these estimators is substantially more efficient than estimators that ignore the ties. As part of our comparison of estimators, we develop new results about models for ties in rankings. We also show that there are settings where, to achieve more efficient estimation, ties should be declared not just when the ranker is actually unsure about how units rank, but also when the ranker is sure about the ranking, but believes that the units are close.

Keywords

Imperfect rankings Isotonic estimation Judgment post-stratification Ranked-set sampling Ties in rankings 

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References

  1. Bohn LL, Wolfe DA (1994) The effect of imperfect judgment rankings on properties of procedures based on the ranked-set samples analog of the Mann-Whitney-Wilcoxon statistic. J Am Stat Assoc 89: 168–176CrossRefGoogle Scholar
  2. Chen H, Stasny EA, Wolfe DA (2005) Ranked set sampling for efficient estimation of a population proportion. Stat Med 24: 3319–3329PubMedCrossRefGoogle Scholar
  3. Dell TR, Clutter JL (1972) Ranked-set sampling theory with order statistics background. Biometrics 28: 545–555CrossRefGoogle Scholar
  4. Fligner MA, MacEachern SN (2006) Nonparametric two-sample methods for ranked-set sample data. J Am Stat Assoc 101: 1107–1118CrossRefGoogle Scholar
  5. Frey J (2007) New imperfect rankings models for ranked set sampling. J Stat Plan Inference 137: 1433–1445CrossRefGoogle Scholar
  6. Frey J (2011) A note on ranked-set sampling using a covariate. J Stat Plan Inference 141: 809–816CrossRefGoogle Scholar
  7. Halls LK, Dell TR (1966) Trial of ranked-set sampling for forage yields. For Sci 12: 22–26Google Scholar
  8. Kvam PH (2003) Ranked set sampling based on binary water quality data with covariates. J Agricult Biol Environ Stat 8: 271–279CrossRefGoogle Scholar
  9. MacEachern SN, Stasny EA, Wolfe DA (2004) Judgement post-stratification with imprecise rankings. Biometrics 60: 207–215PubMedCrossRefGoogle Scholar
  10. McIntyre GA (1952) A method for unbiased selective sampling, using ranked sets. Aust J Agricult Res 3: 385–390CrossRefGoogle Scholar
  11. McIntyre GA (2005) A method for unbiased selective sampling, using ranked sets. Am Stat 59:230–232 (originally appeared in Aust J Agricult Res 3:385–390)Google Scholar
  12. Ozturk O (2007) Statistical inference under a stochastic ordering constraint in ranked set sampling. J Nonparametr Stat 19: 131–144CrossRefGoogle Scholar
  13. Ozturk O (2011) Sampling from partially rank-ordered sets. Environ Ecol Stat 18: 757–779CrossRefGoogle Scholar
  14. Patil GP (1995) Editorial: ranked set sampling. Environ Ecol Stat 2: 271–285CrossRefGoogle Scholar
  15. Robertson T, Wright FT, Dykstra RL (1988) Order restricted statistical inference. Wiley, New YorkGoogle Scholar
  16. Wang X, Lim J, Stokes SL (2008) A nonparametric mean estimator for judgment post-stratified data. Biometrics 64: 355–363PubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsVillanova UniversityVillanovaUSA

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